On commutativity and finiteness in groups

Bull Braz Math Soc, New Series 40(2), 149-180 © 2009, Sociedade Brasileira de Matemática On commutativity and finiteness in groups Ricardo N. Oliveira and Said N. Sidki Abstract. The second author introduced notions of weak permutablity and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite subgroups. Two groups H, K weakly commute provided there exists a bijection f : H → K which fixes the identity and such that h commutes with its image h f for all h ∈ H . The present paper gives support to conjectures about the nilpotency of groups generated by two weakly commuting finite abelian groups H, K . Keywords: commutativity graph, finiteness, weak commutativity, double cosets, nilpotency class, solvablity degree, group extensions. Mathematical subject classification: Primary: 20F05, 20D15; Secondary: 2F18, 20B40. 1 Introduction Commutativity in a group can be depicted by its commutativity graph which has for vertices the elements of the group and where two elements are joined by an edge provided they commute. Greater understanding of the commutativity graph of finite groups has been achieved in recent years and this has had important applications; see for example [5]. For a nontrivial finite p-group, it is an elementary and fundamental fact that it has a non-trivial center and therefore each element in its commutativity graph is connected to every element in the center. Now, suppose a finite group G contains a non-trivial p-group A such that every p-element in G commutes with some non-trivial element in A. Does it follow that G contains a non-trivial normal p-subgroup? In 1976, the second author proved in [6] results along such lines for p = 2 and formulated the following conjecture: if a finite group G contains a non-trivial elementary abelian 2-group A such that every involution in G commutes with some involution from A then A ∩ O2 (G) is non-trivial. Received 3 October 2008. 150 RICARDO N. OLIVEIRA and SAID N. SIDKI This was settled in 2006 by Aschbacher, Guralnick and Segev in [1]. The proof made significant use of the classification theorem of finite simple groups. It was also shown that the result applies to the Quillen conjecture from 1978 about the Quillen complex at the prime 2; see [4]. A configuration which arises in this context is one where the commutation between elements of A and those of one of its conjugates B = A g is defined by a bijection. An approach which we had taken in 1980 to such a weak form of commutativity or permutability was through combinatorial group theory. The following finiteness criterion was proven in [7]. Theorem 1. Let H, K be finite groups having equal orders n and let f : H → K be a bijection which fixes the identity. Then for any two maps a : H → K , b : H → H , the group   G (H, K ; f, a, b) = H, K | hh f = h a h b for all h ∈ H is finite of order at most n exp(n − 1), where h f , h a , h b denote the images of h under the maps f, a, b, respectively. It is to be noted that the proof uses the same argument as that given by I.N. Sanov for the local finiteness of groups of exponent 4; see [8]. The notion of weak commutativity between H and K by way of a bijection is formalized by the group   G (H, K ; f ) = H, K | hh f = h f h for all h ∈ H . When H is isomorphic to K , we simplify the notation to G (H ; f ). If f itself is an isomorphism from H onto K then G (H ; f ) is the same group as χ (H ) and f as ψ in [7]. In addition to finiteness, the operator χ preserves a number of other group properties such as being a finite p-group. Indeed, it was shown later in [3] that more generally, if H is finitely generated nilpotent then so is χ (H ). We are guided in this paper by the following conjecture. Conjecture 1. Let H and K be finite nilpotent groups of equal order and f : H # → K # a bijection. Then G (H, K ; f ) is also nilpotent. The construction G (H, K ; f ) lends itself well to extensions of groups, as , K  be groups having normal subgroups M, N respectively and follows. Let H   H K let M = H , N = K . Let f : H → K , α : M → N be bijections both fixing e. → K  Then, f and α can be extended in a natural manner to a bijection f ∗ : H Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS 151   =G H , K ; f ∗ modulo the normal closure V of M, N  is fixing e such that G isomorphic to G (H, K ; f ). We use this process to produce an ascending chain of groups of G (H, K ; f ) type. For central extensions, we prove Theorem 2. Maintain the above notation. Suppose M, N are central sub, K  respectively and that G (M, N ; α) is abelian. Then, V is an groups of H    2 is central in G.  abelian group. If furthermore H, K are abelian then V, G Although G (H, K ; f ) is finite for finite groups, H, K , since f in general does not behave well with respect to inductive arguments, methods from finite group theory are difficult to apply. At the present stage, we have stayed close to the case where the groups H and K are isomorphic finite abelian groups and more specially to elementary abelian p -groups A p,k of rank k. A step in the direction of proving nilpotency is Theorem 3. Suppose A, B are finite abelian groups of equal order n and let G = G (A, B; f ). Then, the metabelian quotient group GG′′ is nilpotent of class at most n. The next lemma is a natural first step in classifying the groups G (H, K ; f ) for a fixed pair (H, K ). Lemma 1. Let a ∈ Aut (H ) , b ∈ Aut (K ), and g = a f b. Then, the extension −1 γ : G (H, K ; f ) → G (H, K ; g) of h → h a , k → k b is an isomorphism from G (H, K ; f ) onto G (H, K ; g). Let H and K be isomorphic groups by t : H → K . Then in the above lemma, f = f ′ t, g = g ′ t, b = t −1 b′ t   where f ′ , g ′ ∈ Sym H # and b′ ∈ Aut (H ); here H # denotes H \ {e} Therefore, g ′ = a f ′ b′ is an element of the double coset Aut (H ) f ′ Aut (H ). Thus, in order to classify G (H; f )one is obliged to determine the double coset decomposition Aut (H ) \Sym H # /Aut (H ). When the context is clear, we refer to f simply by its factor f ′ . Computations by GAP [2] produce the following data for abelian groups of small rank: for A2,3 , S L(3, 2)\Sym (7) /S L(3, 2) has 4 double cosets; for A2,4 , S L(4, 2)\Sym (15) /SL(4, 2) has 3374 double cosets; for A3,3 , P G L (3, 3) \Sym (13) /P G L (3, 3) has 252 double cosets. Bull Braz Math Soc, Vol. 40, N. 2, 2009 152 RICARDO N. OLIVEIRA and SAID N. SIDKI   The groups G A p,k ; f have the following nilpotency classes and derived lengths. Theorem 4. (i) Let A = A2,k . If k = 3, 4 then G (A; f ) is a 2-group of order at most k 22 +k−1 , has class at most 5 and derived length at most 3; (ii) Let A = A3,3 . Then G (A; f ) is a 3-group of order at most 39 and has nilpotency class at most 2. We discuss a number of other issues. These include the reduction of the number of relations in G (A; f ), finding a bijection equivalent to f which is “closer” to being an isomorphism and also concerning f inducing a bijection between the nontrivial cyclic subgroups of A. The paper ends with three general examples. The first is G (A, f ) where A is a field, seen as an additive group, and where f corresponds to the multiplicative inverse. The second example illustrates the construction of extensions of G = G (A, f ) which are of the same type as G; this produces metabelian  2 k groups having the same order 22 +k−1 and nilpotency classk as χ  A2,k , but not isomorphic to the latter group. The third is G = G A2,k , f where f corresponds to a transposition of A#2,k . 2 Extensions of groups , K  be groups having normal subgroups M, N and let H be a transverLet H  with e ∈ H . Similarly, let K be a transversal of N in K  with sal of M in H /M and K with the quotient e ∈ K . We identify H with the quotient group H  group K /N . Let f : H → K , α : M → N be bijections both fixing e. Given → K  by a bijection γ : M → N (not necessarily fixing e), define f ∗ : H f ∗ : m → m α , f ∗ : mh → m γ h f if h = e. Then f ∗ is a bijection which fixes e and G = G (H, K ; f ) is an epimorphic  = G H , K ; f ∗ . The natural epimorphisms H  → H, K → K image of G  → G having for kernel V , the normal closure of extend to an epimorphism G  M, N  in G. For any group L let ν L denote the exponent of L and γ i (L) the i-th term of the lower central series of L. Define δ : M → N , ε : M → M by mδ = mα mm αγ Bull Braz Math Soc, Vol. 40, N. 2, 2009 −1 γ −1 , m ε = m (m α m γ )γ −1 −1 . ON COMMUTATIVITY AND FINITENESS IN GROUPS 153 Clearly, m δ = e if and only if m = e and similarly, m ε = e if and only if m = e. Theorem 5. Maintain the previous notation. Suppose M, N are central sub, K , respectively and that G (M, N ; α) is abelian. Then, V is an groups of H abelian group such that    , V = M + N + V, G        = M δ, H = N ε, H , V, G      i−1  = V, i H 2 V, γ i H for all i ≥ 1 and Both sets  m δ1 −1  m δ2 −1   ν [V,G] | gcd ν M , ν N , ν H , ν K . (m 1 m 2 )δ | m 1 , m 2 ∈ M , m −1  δ α−1 m ε |m∈M    Furthermore, if H, K are abelian then V, G  2 is central in G.  are central in G. Proof. (1) Let m = e = h. Then,       −1 −1 m, h f = m, m α h f = m αγ h m, m α h f = mm αγ h, m α h f   −1 −1 γ −1 = h, mm αγ hf mm αγ mα h f    −1 h, m δ = h, m δ . mm αγ = In the same manner,         h, m α = mh, m α = mh, m γ h f .m α = mh, m α m γ h f   −1 α γ γ −1 α γ f = mh, m m h (m m ) h     = mε, mαmγ h f = mε, h f .  α  α−1     , we obtain Since h, m δ = m, h f and m δ = m ′ for m ′ = m δ  ′ ε f    ′ α  m , h = h, m ,        δ α−1 ε f , h = h, m δ = m, h f ; m Bull Braz Math Soc, Vol. 40, N. 2, 2009 154 RICARDO N. OLIVEIRA and SAID N. SIDKI  δ α−1 m ε  commutes with K and is therefore central in G.       (2) We obtain from m, h f = h, m δ that the sets [m, K ] , H, m δ are equal and H, K normalize the subgroup [m, K ]. Also, any m ′ ∈ M commutes      with h, m δ = m, h f ∈ [M, K ]. Therefore, M K = M K is abelian.   Since [M, K ] ≤ H, M δ ≤ [N , H ] and [H, N ] ≤ [M ε , K ] ≤ [M, K ] we get     [M, K ] = H, M δ = [H, N ] = M ε , K . that is, m −1 Therefore V is abelian and   V = M + N + Mδ, H ,      = M δ , H , ν V,G | gcd (ν M , ν N ) . V, G [ ] (3) Let m 1 , m 2 ∈ M. Then,     m 1 m 2 , h f = h, (m 1 m 2 )δ ,    m   m1m2, h f = m1, h f 2 m2, h f    = m1, h f m2, h f    = h, (m 1 )δ h, (m 2 )δ , and Thus,      h, (m 1 m 2 )δ = h, (m 1 )δ h, (m 2 )δ .  m δ1  −1  δ −1 m2 (m 1 m 2 )δ | m 1 , m 2 ∈ M  and likewise for is central in G      −1 −1 m ε1 m ε2 (m 1 m 2 )ε | m 1 , m 2 ∈ M .   (4) Since any h ∈ H centralizes M, h f , we have for all m ∈ M,     m, h f , h = e = m δ , h, h and so,    i   m δ , h, h = e, m δ , h i = m δ , h for all v ∈ V and all integers i. Therefore, ν [V,G] | gcd (ν N , ν H ) , gcd (ν M , ν K ) . Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS 155 We calculate for h 1 , h 2 ∈ H −1  δ   m , (h 1 h 2 )−1 = m δ , h 1 h 2 ,  δ −1  δ −1 −1  m , h2 h1 = m , h1h2 ,  δ −1   δ −1 h −1 m , h2 1 m , h1 =  mδ , h2  mδ , h1 h 2 −1 , h  h −1   δ  m , h1 mδ , h2 1 = mδ , h2 mδ , h1 2 ,   δ −1  δ h −1 −1  δ h m , h2 m , h2 1 = mδ , h1 m , h1 2 ,     δ = mδ , h1, h2 . m , h 2 , h −1 1 Therefore,     δ −1 −1   δ −1  δ = m , h 1 , h 2 = m , h 1 , h −1 m , h 1 , h 2 = m δ , h 2 , h −1 , 1 2  δ −1   δ = m , h 1 , h −1 , m , h1, h2 2 −1   δ    δ = m , h2, h1 . m , h 1 , h 2 = m δ , h 2 , h −1 1 Calculate further  δ    h     m , h1h2 = mδ , h2 mδ , h1 2 = mδ , h2 mδ , h1 mδ , h1, h2 ,       δ m , h1h2, h1h2 = mδ , h2, h1h2 mδ , h1, h1h2 mδ , h1, h2, h1h2     = mδ , h2, h1 mδ , h2, h1, h2 mδ , h1, h2      δ m , h1, h2, h2 mδ , h1, h2, h1 . mδ , h1, h2, h1, h2 h  = m δ , h 1 , h 2 , h 1 2 = e. We conclude Thus,   δ m , h 2 , h 1 , h 1 = e. [v, h, h] = e for all v ∈ V . When V is written additively, the action of h on V can be expressed as v (−1 + h)2 = 0. From this we derive the following formulae for the action of H on V : Bull Braz Math Soc, Vol. 40, N. 2, 2009 156 RICARDO N. OLIVEIRA and SAID N. SIDKI for h 1 , h 2 , . . . , h i ∈ H , h2h1 −1 h1 h2h1 = −2 + 2h 1 + 2h 2 − h 1 h 2 , = −2 + 2h 1 + 3h 2 − 2h 1 h 2 , −1 + [h 2 , h 1 ] = −2 (−1 + h 1 ) (−1 + h 2 ) , −1 + [h i , . . . , h 2 , h 1 ] = (−2)i−1 (−1 + h 1 ) (−1 + h 2 ) . . . (−1 + h i ) . In other words, i−1 [v, [h i , . . . , h 2 , h 1 ]] = [v, h 1 , h 2 , . . . , h i ](−2)      i−1  2 . V, γ i H = V, i H , (5) Suppose H, K abelian. Then   [v, [h 1 , h 2 ]] = [v, h 1 , h 2 ]2 = [v, h 1 ]2 , h 2 = e.      = [V, H ] = [V, K ], we conclude that V, G  2 is central. As V, G  , K  are finite groups and let M = Theorem 6. Suppose in the above, H , K  respectively, each of prime m , N = n be cyclic central subgroups of H  G order p. Then, M, N  is an elementary abelian p -subgroup of rank at most |H | + 1. Proof. We have [M, N ] = {e}, [M, K ] = [H, N ] elementary p-abelian subgroup and M, N centralize [M, K ]. Therefore,      M, N G = M, N G = m K , n = m, n H is an elementary abelian p-subgroup of rank at most |K | + 1.  3 Metabelian quotients of G (A, B, f ) It would be interesting to resolve the question of nilpotency of the solvable quotients of G (H, K ; f ). In the next result we consider metabelian quotients of G (A, B; f ). Theorem 7. Suppose A, B are finite abelian groups of equal order n and let G = G (A, B; f ). Then, the metabelian quotient group G/G ′′ is nilpotent of class at most n. Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS Proof. 157 In a metabelian group M, if u ∈ M ′ and xi ∈ M (1 ≤ i ≤ k) then     u, x1 , x2 , . . . , xk = u, xi1 , xi2 , . . . , xik for any permutation (i 1 , i 2 . . . , i k ) of {1, 2, . . . , k}. We have from the relations of G,   −1  −1 b = a. b f [a, b] = a, a f −1  ,b . f −1 Let a ∈ A, b1 , b2 ∈ B. As a f and b1 commute with both a, b1 , it follows that  −1  .b2 [a, b1 , b2 ] = a, b1 , a f   −1 f −1 , b1 , b2 = a. b1     f −1 = −1 a, b1 , a. b1  f −1 We observe that if b1 = b2 = a , then b1 = a. b1 f f b2 . −1 f b2 . Now, we will work in G modulo G ′′ . Let a ∈ A, bi (2 ≤ i ≤ k) ∈ B. From Witt’s formula, as A, B are abelian, we have     a, b2 , b1 = a, b1 , b2 and more generally,     a, b1 , b2 , . . . , bk = a, bi1 , bi2 , . . . , bik for any permutation (i 1 , i 2 . . . , i k ) of {1, 2, . . . , k}. Therefore, if     x2 , . . . , xk = b2 , . . . , bs , as+1 , . . . , ak with b2 , . . . , bs ∈ B and as+1 , . . . , as+k ∈ A then     a, b1 , x2 , . . . , xk = a, b1 , b2 , . . . , bs , as+1 , . . . , ak . Let again a ∈ A, bi (2 ≤ i ≤ k) ∈ B. Suppose that b1 , b2 , . . . , bk−1 , a f are distinct. Bull Braz Math Soc, Vol. 40, N. 2, 2009 158 Then, RICARDO N. OLIVEIRA and SAID N. SIDKI  f −1 bk , a. b1 −1 f  f −1 bk , . . . , a. bk−1 −1 f bk are k distinct elements of B. Suppose further that bk = b j for some 1 ≤ j ≤ k − 1. Then, for some i,   −1 f   f −1 ′ b j ∈ b1 , b2 , . . . , bk−1 . bi j = a. bi In this manner, we have     a, b1 , b2 , .., bk−1 , bk = a, b1 , b2 , . . . , bk−1 , b j   = a, bi , b j , bl , . . . , bm   = a, bi , bi′ j , bl , . . . , bm   = a, b1 , b2 , . . . , bk−1 , bi′ j and b1 , b2 , . . . , bi′ j are distinct. If k = n then [a, b1 , b2 , . . . , bn ] = e and G is nilpotent of class at most n.  The limit n obtained in the proof is clearly too large, especially when compared with available results. Determining the nilpotency degree seems to stem from a more general problem which can be formulated for commutative rings. Problem 1. Let A be a free abelian group of rank m generated by a1 , . . . , am and let A act on a torsion-free Z-module V . Let n be a natural number. Define ⎧ l l ⎫ ⎪ ai11 ai22 . . . ailss ais +1 | 0 ≤ s ≤ m − 1, ⎪ ⎪ ⎪ ⎨ ⎬ S (m, n) = . 1 ≤ i 1 < i 2 < · · · < i s < m, ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 1 ≤ li ≤ n − 1 This set corresponds to a choice of a generator for each of the different cyclic  subgroups of order n in A/An . For instance, S (2, 3) = a1 , a2 , a1 a2 , a12 a2 . Let f be a permutation of S (m, n) and define     U (m, n; f ) = (1 − x) 1 − x f | x ∈ S (m, n) . Suppose that A acts on V such that U (m, n; f ) = {0}. Prove that the action of A on V is nilpotent. Moreover, that it has nilpotency degree at most 3 for n = 2 and degree at most 2 for n ≥ 3. The small bounds have been confirmed by a number of examples using the Groebner basis applied to Q[a1 , a2 , . . . , an ] modulo the ideal generated by U (m, n; f ). Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS 159 4 Reduction of the presentation of χ (A) It is interesting to reduce the number of relations in the definition of G (A, f ), particularly for the sake of applications. This is difficult to carry out in general. We treat here the question for the group χ (A). Given a generating set S of A then define     χ (A, S; m) = A, Aψ | a, a ψ = e for all a ∈ ∪1≤i≤m S i . The construction χ (A, S; 1) does not conserve finiteness in general. For let ψ A = A2,2 be generated by S = {a1 , a2 } . Then, on defining x1 = a1 a2 , x2 = ψ a1 a2 , we find χ (A, S; 1) = x1 , x2  A, where x1 , x2  is free abelian of rank 2. We start with Proposition 1. Let H be a group generated by x 1 , x2 , y1 , y2 such that   y1 , x1 = e = y2 , x2 . Then   y    (i) y1 y2 , x1 x2 = y1 2 , x2 y2 , x1x2 ;     (ii) if in addition x1 , x2 = y1 , y2 = e holds then      y1 y2 , x1 x2 = y1 , x2 y2 , x1 (*);   (iii) if furthermore y1 y2 , x1 x2 = eholds then  H is nilpotent of class at most 2 with derived subgroup H ′ = y1 , x2 . Proof. The first two items are shown directly. The last item follows from   x      y1 , x2 = x1 , y2 , y1 , x2 1 = y1 , x2 , x x     y1 , x2 2 = x1 , y2 2 = x1 , y2 .  Corollary 1. Let A be an abelian group generated by S = {a1 , a2 } and let G = χ (A, S; 2). Then, G = χ (A). For abelian groups A of rank 3, the situation becomes less simple. Proposition 2. Let A be an abelian group generated by S = ai |1 ≤ i ≤ 3. Then the following equations hold in G = χ (A, S; 2): a2 a2ψ  ψ ψ ψ ψ ψ a3 , a1 , ξ = a1 a 2 a3 , a 1 a2 a 3 = a 1 , a3 ψ a1 , a32 Bull Braz Math Soc, Vol. 40, N. 2, 2009 [a2 ,ψ]  ψ = a1 , a32 . 160 RICARDO N. OLIVEIRA and SAID N. SIDKI ψ ψ ψ Proof. On substituting x1 = a1 a2 , x2 = a3 , y1 = a1 a2 , y2 = a3 in (*) of the previous proposition, we obtain  ψ ψ ψ ξ = a1 a 2 a3 , a1 a 2 a 3 = ψ a1 , a3 = ψ a1 , a 3 = ψ a1 , a 3 a2ψ a2ψ a2ψ ψ a 2 , a3 ψ a 2 , a3 ψ a 3 , a1   ψ a3 , a 2 ψ a3 , a2 a2  ψ a 3 , a1  ψ a3 , a1 a2 a2 . Substitute a1 ↔ a3 , a2 ↔ a2 , ψ → ψ above to obtain ξ= ψ Therefore, since a1 , a3 2, we have ψ a1 , a3 ψ a1 , a3 a2ψ a2ψ  a2 a2ψ a 1 , a3 = a3 , a1 = a3 , a1 = a1 , a3 ψ a3 , a1 ψ . !  ψ ψ ψ = a1 , a3 , and a1 , a3 , a1 , a3 has class at most ψ a3 , a1 ψ a3 , a1 ψ a1 , a3 a2 a2 2a2ψ  ψ a1 , a32 = [a2 ,ψ] ψ a1 , a32 = ψ a2 ψ ψ ψ ψ a1 , a32 a2ψ a2ψ 2a2 ψ a1 , a3 ψ a1 , a3 a2 a2 , , , a2  ψ a1 , a32 . Suppose A has odd order. Then, [a2 ,ψ]  ψ ψ a1 , a3 = a 1 , a3  ψ ψ ψ and therefore ξ = a1 a2 a3 , a1 a2 a3 = e. By the previous corollary, we can substitute the ai ’s by their powers in this last equation.  For groups A of odd order the reduction is drastic. Corollary 2. Let A be a finite abelian group of odd order generated by S and G = χ (A, S; 2). Then, G = χ (A). Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS 161 Proof. Let |S| = m ≥ 3. We proceed by induction on m. By the previous proposition, χ (A; 2) = χ (A; 3). We assume χ (A; 2) = χ (A; m − 1). Then we simply apply our argument to the set.   a1 , a2 , . . . , am−2, am−1 am with m − 1 elements and obtain  ψ ψ ψ a1 a2 . . . am−2 (am−1 am )ψ , a1 a2 . . . am−2 (am−1 am ) = e.  Example 1. The following example provides us with a glimpse into the prob  lem of reduction of the of G A p,k , f in general and how it com  presentation pares with that of χ A p,k . Let A, B be isomorphic to A p,3 with respective generators {a1 , a2 , a3 }, {b1 , b2 , b3 }. Define # "     A, B | ai , bi = e i = 1, 2, 3 , . G=       a1 a2 , b1 b2−1 = a1 a3 , b1 b3 = a2 a3 , b2 b3 = e With the use of GAP, we find that the resulting group for p = 3, 5, 7 to be finite metabelian of order p 11 and of nilpotency class 3. We also find that       −1 a1 a2 , b1 b2 = a1 a3−1 , b1 b3−1 = a2 a3−1 , b2 b3−1 = e hold but  j a1 a2 a3 , b1i b2 b3 = e for any 1 ≤ i, j ≤ p − 1.   These results should be compared with those for χ A p,3 which has order p 9 and nilpotency class 2.   We go back to the case χ A p,3 for p = 2. Theorem 8. Let A2,3 be generated by      ψ ψ ψ  S = a1 , a2 , a3 , G = χ A2,3 , S; 2 and ξ = a1 a2 a3 , a1 a2 a3 .   Then the kernel K of the epimorphism φ : G → χ A2,3 extended from ai →  ψ ψ  ai , ai → ai i = 1, 2, 3 is the normal closure of ξ  in G and is free abelian of rank 4. Bull Braz Math Soc, Vol. 40, N. 2, 2009 162 RICARDO N. OLIVEIRA and SAID N. SIDKI Proof. We will show that K is freely generated by   ξ , ξ ai (i = 1, 2, 3) and that G acts on it as follows: for {i, j, k} = {1, 2, 3}, ψ ψ ξ ψ = ξ −1 , ξ ai = ξ −ai , ξ ai a j = ξ −ak = ξ a j ai . We sketch the proof. First, we derive the table ψ a 3 , a2 , a1 ψ ψ a 3 , a2 , a1 ψ a2ψ ψ a3 , a2 , a1 ψ ψ ψ ψ a1 , a2 , a3 a2 , a1 , a3 a2 = ψ a2 , a 1 , a 3 ψ ψ = a2 , a1 , a3 a2 = a2 , a 1 , a 3 = a3 , a 1 , a 2 a2 = a3 , a 2 , a 1 a1 ψ −1 ψ ψ ψ ψ ψ , −1 ψ a 3 , a2 , a1 ψ a3ψ ψ , a 3 , a 2 , a1 −1 , a 3 , a2 , a1 −1 . −1 ψ ψ ψ ψ , a 1 , a3 , a2 a3 a3 a1 −1 ψ = a1 , a3 , a2 ψ ψ = a1 , a3 , a2 ψ ψ = a1 , a3 , a2 ψ ψ = a2 , a 1 , a3 , −1 −1 −1 , , , From this table we conclude that the subgroup generated by $ %   a1ψ  ψ ψ ψ ψ a 3 , a 2 , a 1 , a1 , a 3 , a 2 , a 2 , a 1 , a 3 , a3 , a 2 , a 1 is abelian and normal in G. Next, we find ξ = =    ψ ψ ψ ψ ψ ψ a3 , a1 , a2 a 1 a2 a3 , a 1 a2 a 3 = a2 , a3 , a1   ψ ψ ψ a1 , a 3 , a 2 a2 , a1 , a3 and by permuting the ai ’s, the following holds     ψ ψ ψ ψ ψ ψ ξ = a3 , a2 , a 1 a2 , a 1 , a 3 = a1 , a 3 , a2 a 3 , a2 , a1   ψ ψ ψ = a1 , a2 , a 3 a2 , a3 , a1 . It is straightforward to obtain the action of G on {ξ , ξ ai (i = 1, 2, 3)}, as described above. Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS 163 Let Z [x, y, z] be the polynomial ring in the variables x, y, z with coefficients from Z. The proof is finished by constructing the group as a subgroup of G L (5, Z [x, y, z]): ⎞ ⎛ ⎞ ⎛ 0 0 1 0 0 0 1 0 0 0 ⎜ 0 0 ⎜ 1 0 0 −1 0 ⎟ 0 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ 0 0 0 ⎟ 0 −1 0 ⎟ , a2 → ⎜ 1 0 a1 → ⎜ 0 0 ⎟, ⎝ 0 −1 0 ⎝ 0 0 −1 0 0 ⎠ 0 0 ⎠ x y −x y 1 x −x y y 1 ⎛ ⎞ ⎛ ⎞ 0 0 0 1 0 0 −1 0 0 0 ⎜ 0 0 −1 0 0 ⎟ ⎜ −1 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ψ ⎜ ⎟ ⎜ 0 0 ⎟, 0 0 −1 0 ⎟ a3 → ⎜ 0 −1 0 a1 → ⎜ 0 ⎟, ⎝ 1 0 ⎝ 0 0 0 0 ⎠ 0 −1 0 0 ⎠ x y y −x 1 w w y y 1 ⎞ ⎛ ⎞ ⎛ 0 0 0 −1 0 0 0 −1 0 0 ⎜ 0 ⎜ 0 0 −1 0 0 ⎟ 0 0 −1 0 ⎟ ⎟ ⎜ ⎟ ⎜ ψ ψ ⎜ ⎟ ⎜ 0 0 ⎟ 0 0 0 ⎟ , a3 → ⎜ 0 −1 0 a2 → ⎜ −1 0 ⎟ ⎝ −1 0 ⎝ 0 −1 0 0 0 0 ⎠ 0 0 ⎠ w y y w 1 w y w y 1 from which we find that ⎛ ⎜ ⎜ ξ →⎜ ⎜ ⎝ where z = 4 (−x − y + w). 1 0 0 0 z 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠  5 Fixing a basis Let A = A p,k be written additively and let Bk be the set of bases of A. We f will show that for any bijection f : A → A fixing 0, the set Bk ∩ Bk is nonempty; indeed, , , , f, ,Bk ∩ Bk , lim = 1. p→∞ |Bk | Bull Braz Math Soc, Vol. 40, N. 2, 2009 164 RICARDO N. OLIVEIRA and SAID N. SIDKI Example 2. (i) The following easy example shows that for p odd, a bijection f : A → A may be linear when restricted to each of the 1 -dimensional subspaces and may also permute Bk , without being a linear transformation. Let A = A p,2 be generated by a1 , a2 and define f : A → A by f : ia1 → ia1 , ia2 → ia2 , i (a1 + ja2 ) → i (a1 − ja2 ) for 0 ≤ i, j ≤ p − 1, j = 0. (ii) A map f is anti-additive provided (x + y) f = x f + y f for all x, y such that 0 ∈ {x, y, x + y}. Let F be a field of characteristic different from 3 and such that its multiplicative group F # does not contain elements of order 3. Then, multiplicative inversion defined by f : 0 → 0, x → x −1 is anti-additive. Concerning the first example, the situation for k ≥ 3 is quite different as can be seen from a result of R. Baer from 1939 ([9], Th. 2, page 35): let G be an abelian p-group such that G contains an element of order p n and contains at least 3 independent elements of such order. Then any projectivity of G onto another abelian group H is induced by an isomorphism. It follows then Lemma 2. Let A = A p,k . Suppose f is a permutation of A# . If f permutes the set Bk of bases of A then f is a linear transformation when k ≥ 3 and when p = 2, k = 2. Proof. As f permutes the set Bk of bases of A, it induces a projectivity on A. The case p = 2, k = 2 is easy.  Definition 1. Let A = A p,k and f be a permutation of A# . A subset C of A of linearly independent elements is said to be f -independent if C f is also a linearly independent set. Let N ( p, j) denote the number of f -independent subsets C of A with |C| = j. Proposition 3. Maintain the previous notation. Then, the following inequality holds for all k ≥ 1,   N ( p, k) ≥ p k − 1 1≤i≤k−1 Bull Braz Math Soc, Vol. 40, N. 2, 2009  p k− j − 1  j p − 2 p j−1 + j + 1 . p−1 ON COMMUTATIVITY AND FINITENESS IN GROUPS Furthermore, lim p→∞ , , , f, ,Bk ∩ Bk , |Bk | 165 = 1. Proof. Clearly, N ( p, 1) = p k − 1. Let  k ≥ 2, C be f -independent and |C| = j ≥ 1. Denote U = C , W = C f . Then, |U | = |W | = p j , , , ,  , #  ,U − C , = ,W # − C f , = p j − 1 − j. p k− j − 1 non-trivial cyclic subgroups in the quop−1 tient group A/U . For each such cyclic subgroup, choose a representative P in A and also choose a generator v for each P. Suppose U = A. There are Fix such a P = v. Then, each element in the set   L = u + iv | u ∈ U, 1 ≤ i ≤ p − 1 , , is independent of C and |L| = , L f , = p j ( p − 1). The elements of L f \W # =   L f \ W # − C f are independent of C f and , , f   , L \W # , ≥ p j ( p − 1) − p j − 1 − j p j+1 − 2 p j + j + 1. = Therefore, N ( p, j + 1) ≥ N ( p, j) and   N ( p, k) ≥ p k − 1 -  p k− j − 1  j+1 p − 2pj + j + 1 p−1 1≤ j≤k−1  p k− j − 1  j+1 p − 2pj + j + 1 . p−1 Finally, since |Bk | = - 0≤ j≤k−1   k p k − p j = p (2) , , , f, N ( p, k) = ,Bk ∩ Bk , , Bull Braz Math Soc, Vol. 40, N. 2, 2009 - 0≤ j≤k−1   p k− j − 1 , 166 RICARDO N. OLIVEIRA and SAID N. SIDKI we conclude , , , f, ,Bk ∩ Bk , and lim p→∞ |Bk | , , , f, ,Bk ∩ Bk , |Bk | ≥ - 1≤ j≤k−1  p j+1 − 2 p j + j + 1 k ( p − 1)k−1 p (2) = 1 follows easily.   Corollary 3. Let f : A#p,k → A#p,k be a bijection. Then there exists g an element in the double coset G L(k, p). f.G L(k, p) and there exists a basis C of A#p,k such that g fixes point-wise the elements of C. 6 Permuting the set of cyclic subgroups The commutation between two elements in a group imply the commutation of the cyclic groups generated by them. For this reason, it is important to consider commutation correspondence between cyclic subgroups. Let A = A p,k and let f be a permutation of A# . Define R ⊂ A#p,k × A#p,k by   R = (a i , b j ) (1 ≤ i, j ≤ p − 1) | a f = b . We will prove that R contains at least p − 1 permutations g of A#p,k such that    i g  i a = a g for all 1 ≤ i ≤ p − 1. Therefore, G A p,k ; f is a quotient of G A p,k ; g for each one of these g’s. For this purpose, we construct a multiedge digraph L from R, having vertices the non-trivial cyclic subgroups Ci of A    i ′  j  ′ j ! ′ and edges C, C whenever C = a , C = a and f : a i → a ′ . Then L is a regular graph, in the sense that there are exactly p − 1 edges coming into and p − 1 edges leaving each vertex.   We enumerate the vertices of L and let N = Ni j be the incidence matrix with respect to this enumeration; that is Ni j = l if and only there is a total of l edges connecting the vertex i to the vertex j. Then N is doubly stochastic, as all row and column sums of N are equal to p − 1. A permutation g contained in R corresponds to a non-zero monomial N1,1σ N2,2σ . . . Nk,k σ for some permutation σ of {1, 2, . . . , k}.     Definition 2. Let M = Mi j , N = Ni j be k × k matrices over the real numbers. Then, (i) M, N are equivalent provided there exist permutational matrices S, T such that M = S N T ; (ii) N is said to be totally singular provided N1,1σ N2,2σ . . . Nk,k σ = 0 Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS 167 for all permutations σ of {1, 2, . . . , k}. Proposition 4. Let N be a totally singular k × k matrix over the real numbers. Then N is equivalent to a matrix which contains a submatrix 0(k−l)×(l+1) for some l ≥ 0. By induction  on k.The cases k = 2, 3 are easy; that is, if k = 2 then ∗ 0 N is equivalent to and the if k = 3 then N is equivalent to one of ∗ 0 ⎞ ⎛ ⎞ ⎛ ∗ ∗ ∗ ∗ ∗ 0 ⎝ ∗ ∗ 0 ⎠, ⎝ ∗ 0 0 ⎠. ∗ 0 0 ∗ ∗ 0 Proof. Suppose that the assertion is true for k. We consider N of dimension k + 1. Then, we can assume that there exist an l ≥ 0 such that ⎛ ⎞ a11 ... U1×(l+1) Bl×(k−l−1) Cl×(l+1) ⎠ . N = ⎝ Vl×1 W(k−l)×1 D(k−l)×(k−l−1) 0(k−l)×(l+1) If U1×(l+1) or any row of Cl×(l+1) is null then we obtain the desired form. We can also assume that U1×(l+1) = (. . . , a1k ), a1k > 0. Therefore, we have the (l + 1) × (l + 1) matrix     . . . a1k U1×(l+1) . = Rl×l Sl×1 Cl×(l+1) We have Y(k−l)×(k−l) = and therefore   W(k−l)×1 D(k−l)×(k−l−1)   Z (l+1)×(l+1) N= . Y(k−l)×(k−l) 0(k−l)×(l+1) Now, one of Y(k−l)×(k−l) , Z (l+1)×(l+1) is totally singular; suppose it is the first one. Then we may assume   ′ ... Ym×(m+1) . Y(k−l)×(k−l) = . . . 0(k−l−m)×(m+1) Hence ⎛ ∗ ∗ ′ Ym×(m+1) Z (l+1)×(l+1) ⎞ ⎠ N = ⎝ ... 0(k−l)×(l+1) . . . 0(k−l−m)×(m+1) and we obtain in N a (k − l − m) × (m + 1 + l + 1) block of zeroes where the sum of the dimensions is k + 1.  Bull Braz Math Soc, Vol. 40, N. 2, 2009 168 RICARDO N. OLIVEIRA and SAID N. SIDKI Corollary 4. Maintain the previous notation. Suppose the entries of N are non-negative. If in addition N is doubly stochastic then N = 0. Let the row sum be s. There exists l ≥ 0 such that   Z l×(l+1) X l×(k−l) N= . Y(k−l)×(k−l) 0(k−l)×(l+1)   Z l×(l+1) is (l + 1) s whereas the row sum Therefore the column sum of 0(k−l)×(l+1) is at most ls; hence s = 0.  Proof. We go back to our graph L and its incidence matrix N which is doubly stochastic with s = p − 1. Then there exists a monomial N1,1σ N2,2σ . . . Nk,k σ = 0 and so Ni,i σ = 0 for all i. This produces for us a bijection g : A#p,k → A#p,k . By removing the edges corresponding to g, the graph L is reduced to one which is (s − 1)-regular. Therefore, we can produce in this manner p − 1 permutations g. Clearly, if f is an isomorphism on the cyclic subgroups then all the permutations g are equal. 7 Classification of G (A; f ) for A of small rank We treat in this section groups G (A, B; f ) were A, B are finite abelian groups generated by at most 4 elements. Proposition 5. Suppose A = a , B = b are cyclic groups having equal finite orders n. Then, G = G (A, B; f ) is isomorphic to A × B. Proof. Suppose G is not abelian. Let 1 < r, s < n be minimal integers such that [a, br ] = e = [a s , b]. Then,   i f : a | 1 < i < n, gcd (i, s) = 1   → b j | 1 ≤ j < n, r | j , n n < ; φ (s) s r   f −1 : bi | 1 < i < n, gcd (i, r ) = 1   → a j | 1 ≤ j < n, s| j , n n φ (r ) < ; r s n n n φ (s) φ (r ) < φ (s) < r s r  which is a contradiction. Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS 169 Example 4. The following example shows that relaxing f from bijection to surjection may not maintain the finiteness of G (A, B; f ). Let A = a de a cyclic group of order p 3 , B = b be cyclic of order p 2 and define f : A → B by choosing surjective maps f : e → e,    p → b p \ {e} , A\ a  p   a \ {e} → B\ b p .   Then the relations a, a f = e in G (A, B; f ) are equivalent to a p , b p  being central in G. Therefore, G/a p , b p  is isomorphic to the free product C p ∗ C p . Proposition 6. Let A = a1 , a2  , B = b1 , b2  be homogenous abelian groups of rank 2, both having finite exponent n. Then G = G(A, B; f ) is nilpotent of class at most 2 and its derived subgroup is cyclic of order divisor of n. Proof. Let us call an element of A which is part of some 2-generating set of A j primitive. The non-primitive elements are of the form is a1i a2 where gcd (i, n) = 1 = gcd ( j, n); therefore, their number is (n − ϕ (n))2 . The number of primitive elements is n 2 − (n − ϕ (n))2 = 2nϕ (n) − ϕ (n)2 . The difference between the number of primitive elements and the non-primitives is positive: 2nϕ (n) − ϕ (n)2 − (n − ϕ (n))2 = 4nϕ (n) − 2ϕ (n)2 − n 2 and   ϕ (n) ϕ (n) 2 ϕ (n) 2 , +2 ≥1+2 n n n since ϕ (n) 1 < 1. ≤ n 2 Since f is a bijection we may suppose  f : a1→ b1 . Now, any 2-generating set of A containing a1 has the form a1 , a1l a2m where gcd (m, n) = 1; there are nϕ (n) such elements a1l a2m . As nϕ (n) > (n − ϕ (n))2 , we may suppose f : a 2 → b2 . j As f : a1i a2 → b1k b2l , we have     j j a1i a2 , b1k b2l = a1i , b2l a2 , b1k = e. Bull Braz Math Soc, Vol. 40, N. 2, 2009 170 RICARDO N. OLIVEIRA and SAID N. SIDKI     j Since a1 , b2 commute with a2 , b1k , while a2 , b1 commute with a1i , b2l , we    j conclude that b1k , a2 = a1i , b2l is central. We note that the size of   j a1i a2 | gcd (i, n) = 1, j = 0 is ϕ (n) (n − 1), whereas the size of   k l b1 b2 | k = 0, gcd (l, n) = 1, l = 0 is (n − 1) (n − ϕ (n) − 1). As the first set is larger than the second, there exist i, j with gcd (i, n) = j 1, j = 0 such that f : a1i a2 → b1k b2l where gcd (l, n) = 1. We rewrite a1i l as a1 and b2 as b2 and conclude that [a1 , b2 ] is central. Similarly, [a2 , b1 ] is also central. Hence,  n  n a1 , b2 = a2 , b1 = e and there exist 0 ≤ s, t ≤ n − 1 such that    s    t a1 , b2 = a2 , b1 , a2 , b1 = a1 , b2     and G ′ = a1 , b2 = a2 , b1 .    7.1 The groups G A p,k ; f for p = 2, 3 and k = 3, 4 We consider the groups       G A2,3 ; f , G A2,4 ; f , G A3,3 ; f , their orders, nilpotency classes c and derived lengths d. We write the group A p,k additively. (i) The group A2,3 is   0, a1 , a2 , a3 , a1 + a2 , a1 + a3 , a2 + a3 , a1 + a2 + a3 , which we enumerate lexicographically and identify its elements with their positions in this order. The group S L(3, 2) in its linear action on A#2,3 , is generated by the permutations (2, 7, 4, 6, 5, 8, 3), (2, 8, 7)(3, 4, 6). Using GAP, we find that there are 4 double cosets in S L(3, 2)\ Sym(7)/S L(3, 2), which are represented by the permutations   () , (6, 7) , (6, 7, 8) , (5, 6, 7, 8) . Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS 171 Each permutation produces for us a bijection f and a group as in the table below f () (6, 7) ,  , ,G A2,3 ; f , (6, 7, 8) (5, 6, 7, 8) c d 2 10 3 2 2 10 3 2 2 8 2 2 2 8 2 2 Further analysis shows that these 4 groups are non-isomorphic. (ii) The group A2,4 is treated in a similar manner. We find that there are 3374 double  cosets in SL(4, 2)\Sym (15) /S L(4, 2) . The corresponding groups G A2,4 ; f have orders 29 , 210 , 211 , 212 , 213 , 215 , 219 . There are 5 representatives f for which the groups have maximum order. We list them below with their invariants c, d: f ,  , ,G A2,4 ; f , c d 2 19 4 2 2 19 3 3 2 19 5 3 (9, 11)(10, 13)(12, 14) 219 5 3 (9, 12)(10, 13)(11, 14) 19 4 2 () (15, 16) (11, 14)(15, 16) 2 Further analysis shows that these 5 groups are non-isomorphic. (iii) The set of 1-dimensional subspaces of A3,3 has size 13. The group G L(3, 3) in its linear action on A3,3 induces the group P G L(3, 3). There are 252 double cosets in P G L(3, 3)\Sym(13)/P G L(3, 3). A double coset representative corresponds to a bijection g of A#3,3 which is linear on the 1-dimensional subspaces of A3,3 . We find that the corresponding groups have orders 36 , 37 , 38 , 39 . The groups of order 36 are clearly isomorphic to A3,3 × A3,3 . Those of higher order have nilpotency class 2. There is a unique group of maximum order 39 which clearly is isomorphic to χ A3,3 . Bull Braz Math Soc, Vol. 40, N. 2, 2009 172 RICARDO N. OLIVEIRA and SAID N. SIDKI 8 Three general examples Working by hand with G = G (H, K ; f ), it is easy to produce many consequences from the defining relations: given h ∈ H, k ∈ K , then the equalities      −1  h, k = h, h f k = k f h, k hold and these serve to define the two maps   −1 α : (h, k) → h, h f k , β : (h, k) → k f h, k on the set H × K . Finding equivalent forms for [h, k] according to the above process corresponds to calculating the orbits of the group α, β in its action on H × K . We will illustrate this sort of analysis in the examples below. 8.1 The multiplicative inverse function Let F, Ḟ be isomorphic fields, via a → ȧ. Define f : 0 → 0, k → k̇ −1 for k = 0 and let  / .  1 = e for a = 0 . G = G (F, inv) = F, Ḟ | a, ȧ   Given an integer m , we have a, mȧ = e. Therefore, if F = G F ( p) or Q, the group G is isomorphic to F × F. We will prove   Theorem 9. Let F = G F 2k where 2k − 1 is a prime number. Then, G (F; inv) is nilpotent of class at most 2. It appears that this result holds more generally. We develop below formulas for general fields F. Lemma 3. Then Let i ≥ 1, b ∈ F, b = 0 and suppose (2i − 1)! is invertible in F. 0 1 2 2i−2 2 1 − 1)!) ((i b , (αβ)i : (0, b) →  2 , (2i − 2)! 2i−1 (i − 1)! b 0  i−1 2 1 2i 2 (i − 1)! 1 − 1)! (2i b . (αβ)i α : (0, b) →  2 , (2i − 1)! 2i−1 (i − 1)! b Bull Braz Math Soc, Vol. 40, N. 2, 2009 (2i − 1)! ON COMMUTATIVITY AND FINITENESS IN GROUPS 173 Let p be an odd prime number. Then, p+1 (αβ) 2   p−1 2 : (0, b) → 0, (−1) b modulo p. Proof. The first formulae can be verified in a straightforward manner. In case the characteristic of F is a prime number p then ⎛ ⎞ 2 p−2 p−1 ! p−3 ! p+1 2 2 b⎠ (αβ) 2 : (0, b) → ⎝0, ( p − 2)! and by Wilson’s theorem, 0  2 1    p − 1 ( p − 1)! = −12 −22 . . . − 2   2 p−1 p−1 = (−1) 2 ! ≡ −1 modulo p, 2   p+1 p−1 2 ! ≡ (−1) 2 modulo p. 2   Lemma 4. Let L = G F ( p) when charac(F) = p and L = Z when charac(F) = 0. Let    T = a, b | a = 0 = b, ab ∈ L . Then, α, β : T → T and for all integers i 1 , . . . , i s , j1 , . . . , js ,   α i1 β j1 . . . α is β js : (a, b) → a ′ , b′ , where a′ = 1≤s≤k (i 1 + · · · + i s + j1 + · · · + js + ab) a, 1≤s≤k (i 1 + · · · + i s + j1 + · · · + js−1 + ab) b′ = 1≤s≤k (i 1 + · · · + i s + j1 + · · · + js−1 + ab) b. ab1≤s≤k−1 (i 1 + · · · + i s + j1 + · · · + js + ab)     The relations α i , β j = α j , β i hold on the set T , for all integers i, j. Bull Braz Math Soc, Vol. 40, N. 2, 2009 174 Proof. RICARDO N. OLIVEIRA and SAID N. SIDKI We calculate only the action of α i1 β j1 :     i1 i 1 + ab αi1 b a, + b = a, (a, b) → ab a   i 1 + ab j1 β j1 b → a, a+ ab i 1 + ab   i 1 + j1 + ab i 1 + ab = a, b . i 1 + ab ab It is direct to check that the first general non-trivial relation happens for k = 3: j1 = −i 1 − i 2 , j2 = i 1 , i 3 = −i 1 − i 2 , j3 = i 2 ; that is, α i1 β −i1 −i2 α i2 β i1 α −i1 −i2 β i2 = e which in turn is equivalent to  i j  j i α ,β = α ,β .  Lemma 6. Let charac (F) = 2, a, b ∈ F such that a = 0 = b, ab = 1. ab Define c = c (a, b) = 1+ab . Then, α 2 = β 2 = e and for all integers k,   : (a, b) → ck a, c−k b ,   (αβ)k β : (a, b) → ck−1 a, c−k b . (αβ)k The orbit of (a, b) under the action of α, β has length 2.o(c). As both   aci ∈ F and ḃċ−i ∈ Ḟ invert w = a, ḃ for all i ≥ 0, we conclude that the      subgroup 1 + ci a, 1 + ċi ḃ | i = 0 centralizes w. If c satisfies a monic polynomial over G F(2), which is the sum of an odd number of monomials,  4 then a and ḃ centralize w and so a ḃ = e. ab is a Let a = 0 = b, ab = 1. Since c = 1+ab    # i generator of the multiplicative group F we have 1 + c a | i ≥ 0 = F #  2   and so, w = a ḃ = a, ḃ is central in G. Therefore, G is nilpotent of class at most 2. Proof of Theorem 10. Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS 175 We obtain by using GAP:   for F = G F 23 , the group G has order 28 and nilpotency class 2;   for F = G F 24 the group G has order 211 and nilpotency class 2;   for F = G F 33 the group G is abelian, isomorphic to F × F. Example 5. 8.2 Group extension of χ (A)   Let A, B be groups isomorphic to A2,k , k ≥ 3, generated by {ai | 1 ≤ i ≤ k}, {bi | 1 ≤ i ≤ k}, respectively. Define A = ai | 2 ≤ i ≤ k , B = bi | 2 ≤ i ≤ k  and let f : A → B be the isomorphism extended from the map ai → bi 2 ≤      B are central i ≤ k . Then, G(A, B; f ) is isomorphic to χ A2,k−1 . Both A, ∗   extensions of A, B, respectively. Define f : A → B by a1 → b1 , h → b1 h f , a1 h → h f for h ∈ A# ; this corresponds to choosing the bijections α : e → e, a1 → b1 , γ : e → b1 , a1 → e.     We sketch below a proof that the group G = G A, B; f ∗ is metabelian, 2k +k−1 has order and has nilpotency class k. Yet, G is not isomorphic to  2 χ A2,k : for whereas the commutator subgroup of χ A2,k is of exponent 2,   f G ′ is of exponent 4. Indeed, G ′ is generated by a j , a1 for j > 1 (their number is k − 1, each of order 2) and by  f  a j1 , a j2 , a j3 , . . . , a jr where j1 > j2 > . . . > jr > 1 (their number is 2k−1 − k, each of order 4). We develop the proof in steps. →  Let t : A B be the natural isomorphism extended from ai → bi .     partitions under the substitutions α, β into (1) The set x, y t | x, y ∈ H three types of orbits. Let u = w ∈ A. Then orbits types are as follows:             a1 , u t , u, u t , u, b1 , a1 u, b1 , a1 u, b1 u t , a1 , b1 u t ;             (ii) u, w t , a1 uw, w t , a1 uw, u t , w, u t , w, b1 u t wt , u, b1 u t wt ; (i) Bull Braz Math Soc, Vol. 40, N. 2, 2009 176 RICARDO N. OLIVEIRA and SAID N. SIDKI           (iii)  a1 u, b1 wt  , a1 uw, b1 wt , a1 uw, b1 u t , a1 w, b1 u t , a1 w, b1 u t wt , a1 u, b1 u t wt .  (2) The following equalities hold for all x, y, z ∈ A: −1          t   , z, y t , x = y, x t , z . x, y t = y, x t , x , y, z t = z, y t , x     In each orbit we find x, y t , y, x t . Proof of (2).  We conclude that for all x, y, z ∈ A,         x, (yz)t = x, (zy)t = y, x t z, x t x, z t , y t ,         x, (yz)t = yz, x t = y, x t y, x t , z z, x t ,       z, x t x, z t , y t = y, x t , z z, x t ,  x, z t , y t [x t ,z ]   = y, x t , z ,  [z t ,x ]  = y, x t , z , x, z t , y t  t  t    y , z , x = y, x t , z ,   and  −1   z t , x, y t = y, x t , z   −1  −1  t   −1 z t , x, y t = y, x t , z = x, y t , z = x , y, z t = z, y t , x .    (3) We will show that a1 inverts u, w t for all u, w ∈ A. Proof of (3).   Clearly a1 inverts a1 , w t . In the calculations below, given a word ∗ ∗ x ∗ ∗ we introduce dots around ∗ ∗  x as ∗ ∗ .x. ∗ ∗ indicating that x will be substituted by x f x x f , if x ∈ A, −1 −1 ∗ ∗ f f or by x ( ) x x ( ) , if x ∈  B. Let u, w ∈ A. Then,  a u, w t 1 = (a1 u) .w t .uwt a1 = (uw) wt . (a1 uw) .w t a1 u  = (uw) u t (a1 uw) .u t .a1 = (uw) u t wu t u = wu t wu t −1 u  u   . = w, u t = u, w t = u, w t Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS Furthermore,  a   a1 u, b1 wt 1 = u.b1 wt .a1 ub1 wt a1 = uw b1 wt .a1 uw.b1 wt a1       = uw b1 u t (a1 uw) .b1 u t .a1 = uw b1 u t .a1 w. b1 u t a1 u   = uw.b1 u t w t . (a1 w) b1 u t wt a1 u     = b1 u t wt .a1 u. b1 u t wt a1 u       = b1 w t .a1 u. b1 wt a1 u. = b1 wt , a1 u . (4) We claim     y  y  yt zt , x , zt , x = e, z t , x = z t , x and the group G is metabelian. Proof of (4). Then  Apply a1 to        x, (yz)t = x, (zy)t = x, y t x, z t x, z t , y t .      a (yz)t , x = y t , x z t , x x, z t , y t 1    z t  t  z ,x = yt zt , x = yt , x     t  t = y , x y , x, z t z t , x ,  x, z t , y t a1 = = =  x, z t , y  t = = [z t ,x ] y t , x, z t −[z t ,x ]  z, x t , y  t  z , x, y ,  t a z , x, y 1   x, z t , y     x, z t , y t = x, z t , y ,   −1  x, z t , y = x t , z, y t −1  , = z t , x, y t       −1 −1 x, z t , y t . x, z t , yt =  Bull Braz Math Soc, Vol. 40, N. 2, 2009 177 178 RICARDO N. OLIVEIRA and SAID N. SIDKI Thus, and   x, z t yt y  = x, z t yt   x, z t commutes with x, z t . Observe that uv (uv)t  u t v t  uvt   = x, y t = x, y t = x, y t x, y t  [u,vt ]   = x, y t . It follows that G is metabelian. and so, x, y t (4.1) The commutator subgroup is generated by  t w a j , ai where j > i, w ∈ ar | r = 1, i, j . We can improve upon the description of this generating set by using:  t a t a  a j , ai , ak i ak , a tj , ai j = e,  a  ai , a tj , ak a j , akt , ai j = e,    ai , a tj , ak akt , a j , ai = e,  t      a j , ai , ak = akt , a j , ai = a tj , ak , ai       = ait , ak , a j = akt , ai , a j = ait , a j , ak . Therefore the generators have the form  t  a j1 , a j2 , a j3 , . . . , a jr where j1 ≥ j2 ≥ . . . ≥ jr . If j1 = j2 then     t a j1 , a j2 = a tj1 , a1 ,   2  t     a j1 , a j2 , a j3 = a tj1 , a1 , a j3 = a tj1 , a j3 , a1 = a tj1 , a j3 . Thus G ′ is generated by:   t a j , a1 (1 < j ≤ k) (in total of k − 1, each of order dividing 2) and by  t  a j1 , a j2 , a j3 , . . . , a jr for all j1 > j2 > . . . > jr > 1 Bull Braz Math Soc, Vol. 40, N. 2, 2009 ON COMMUTATIVITY AND FINITENESS IN GROUPS 179 , , (in total of 2k−1 − k, each of order dividing 4). Hence, ,G ′ , divides 2k−1 . k−1 k k k k 42 −k = 2k−1+2 −2k = 22 −k−1 and |G| divides 22 −k−1 22k = 22 −1 2k . The nilpotency class of G is at most k and the commutator of highest weight is apparently akt , ak−1 , ak−2 , . . . , a1 . Rather than effecting the final construction, we just remark that computations in GAP confirm the structural information obtained above for k ≤ 5. 8.3 A transposition Let A = A2,k and let f correspond to a transposition. Since S L (k, 2) is 2# # transitive  on A 2,k , any transposition of A2,k is equivalent to f . Detailed analysis of G A2,k ; f indicates that it has the same order as χ (A), but not isomorphic to the latter. Again, this is confirmed by computations in GAP. Acknowledgments. The authors thank Alexander Hulpke for assistance with the double cosets package in GAP. The second author acknowledges support from the Brazilian agencies CNPq and FAPDF. References [1] M. Aschbacher, R. Guralnick and Y. Segev, Elementary Abelian 2-subgroups of Sidki-Type in Finite Groups. Groups Geom. Dyn., 1 (2007), 347–400. [2] The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.4.9 (2006). (http://www.gap-system.org). N. Gupta, N. Rocco and S. Sidki, Diagonal Embeddings of Nilpotent Groups. Illinois J. of Math., 30 (1986), 274–283. [3] [4] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups. Adv. Math., 28 (1978), 101–128. [5] A.S. Rapinchuk, Y. Segev and G.M. Seitz, Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. J. Amer. Math. Soc., 15 (2002), 929–978. S. Sidki, Some Commutation Patterns between Involutions of a Finite Group II. J. of Algebra, 39 (1976), 66–74. [6] [7] [8] [9] S. Sidki, On Weak Permutability between Groups. J. of Algebra, 63 (1980), 186– 225. M. Vaughan-Lee, The Restricted Burnside Problem, Second edition, Oxford Science Publications (1993). M. Suzuki, Structure of a Group and the Structure of its Lattice of Subgroups, Ergebnisse der Math., Springer Verlag (1956). Bull Braz Math Soc, Vol. 40, N. 2, 2009 180 Ricardo N. Oliveira Departamento de Matemática Universidade Federal de Goiás Goiânia, GO BRAZIL E-mail: ricardo@mat.ufg.br Said N. Sidki Departamento de Matemática Universidade de Brasília 70910-900 Brasília, DF BRAZIL E-mail: sidki@mat.unb.br Bull Braz Math Soc, Vol. 40, N. 2, 2009 RICARDO N. OLIVEIRA and SAID N. SIDKI