Journal of Applied Mathematics and Computing, 2008
A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a u... more A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a union of n distinct conjugacy classes of G. In some research papers, the question of finding all positive integer n such that there is an n-decomposable finite group was posed. In this paper, we investigate the structure of 9- and 10-decomposable non-perfect finite groups. We prove that a non-perfect group G is 9-decomposable if and only if G is isomorphic to Aut(PSL(2,32)), Aut(PSL(3,3)), the semi-direct product Z 3∝(Z 5×Z 5) or a non-abelian group of order pq, where p and q are primes and p−1=8q, and also, a non-perfect finite group G is 10-decomposable if and only if G is isomorphic to Aut(PSL(2,17)), PSL(2,25):23, a split extension of PSL(2,25) by Z 2 in ATLAS notation (Conway et al., Atlas of Finite Groups, [1985]), Aut(U 3(3)) or D 38, where D 38 denotes the dihedral group of order 38.
Journal of Applied Mathematics and Computing, 2000
For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is cal... more For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is called n-centralizer if #Cent(G) =n, and primitiven-centralizer if $\# Cent(G){\text{ = \# }}Cent{\text{(}}\frac{G}{{Z(G)}}){\text{ = }}n$ . In this paper we compute the number of distinct centralizers of some finite groups and investigate the structure of finite groups with exactly six distinct centralizers. We prove that ifG is a 6-centralizer group then % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGhbaabaGaamOwaiaacIcacaWGhbGaaiykaaaacaqGGaGaeyyrIaKa% aeiiaGqaciaa-readaWgaaWcbaGaa8hoaaqabaGccaGGSaGaaeiiai% aa-feadaWgaaWcbaGaa8hnaaqabaGccaGGSaGaaeiiaiaabQfadaWg% aaWcbaGaaeOmaaqabaGccaqGGaGaey41aqRaaeiiaiaabQfadaWgaa% WcbaGaaeOmaaqabaGccaqGGaGaey41aqRaaeiiaiaabQfadaWgaaWc% baGaaeOmaaqabaGccaqGGaGaae4BaiaabkhacaqGGaGaaeOwamaaBa% aaleaacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaa% leaacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaale% aacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaaleaa% caqGYaaabeaaaaa!62C4!\[\frac{G}{{Z(G)}}{\text{ }} \cong {\text{ }}D_8 ,{\text{ }}A_4 ,{\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ or Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} \] .
Journal of Applied Mathematics and Computing, 2005
LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is called... more LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is calledn-centralizer if #Cent(G)=n, and primitiven-centralizer if #Cent(G)=#Cent(G/Z(G))=n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and ifG is a finite group such thatG/Z(G)≃A5, then #Cent(G)=22 or 32. Moroever, we prove that A5 is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A5 in terms of the number of centralizers
Journal of Applied Mathematics and Computing, 2008
A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a u... more A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a union of n distinct conjugacy classes of G. In some research papers, the question of finding all positive integer n such that there is an n-decomposable finite group was posed. In this paper, we investigate the structure of 9- and 10-decomposable non-perfect finite groups. We prove that a non-perfect group G is 9-decomposable if and only if G is isomorphic to Aut(PSL(2,32)), Aut(PSL(3,3)), the semi-direct product Z 3∝(Z 5×Z 5) or a non-abelian group of order pq, where p and q are primes and p−1=8q, and also, a non-perfect finite group G is 10-decomposable if and only if G is isomorphic to Aut(PSL(2,17)), PSL(2,25):23, a split extension of PSL(2,25) by Z 2 in ATLAS notation (Conway et al., Atlas of Finite Groups, [1985]), Aut(U 3(3)) or D 38, where D 38 denotes the dihedral group of order 38.
Journal of Applied Mathematics and Computing, 2000
For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is cal... more For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is called n-centralizer if #Cent(G) =n, and primitiven-centralizer if $\# Cent(G){\text{ = \# }}Cent{\text{(}}\frac{G}{{Z(G)}}){\text{ = }}n$ . In this paper we compute the number of distinct centralizers of some finite groups and investigate the structure of finite groups with exactly six distinct centralizers. We prove that ifG is a 6-centralizer group then % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGhbaabaGaamOwaiaacIcacaWGhbGaaiykaaaacaqGGaGaeyyrIaKa% aeiiaGqaciaa-readaWgaaWcbaGaa8hoaaqabaGccaGGSaGaaeiiai% aa-feadaWgaaWcbaGaa8hnaaqabaGccaGGSaGaaeiiaiaabQfadaWg% aaWcbaGaaeOmaaqabaGccaqGGaGaey41aqRaaeiiaiaabQfadaWgaa% WcbaGaaeOmaaqabaGccaqGGaGaey41aqRaaeiiaiaabQfadaWgaaWc% baGaaeOmaaqabaGccaqGGaGaae4BaiaabkhacaqGGaGaaeOwamaaBa% aaleaacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaa% leaacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaale% aacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaaleaa% caqGYaaabeaaaaa!62C4!\[\frac{G}{{Z(G)}}{\text{ }} \cong {\text{ }}D_8 ,{\text{ }}A_4 ,{\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ or Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} \] .
Journal of Applied Mathematics and Computing, 2005
LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is called... more LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is calledn-centralizer if #Cent(G)=n, and primitiven-centralizer if #Cent(G)=#Cent(G/Z(G))=n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and ifG is a finite group such thatG/Z(G)≃A5, then #Cent(G)=22 or 32. Moroever, we prove that A5 is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A5 in terms of the number of centralizers
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