Finite Groups and Representation Theory

The basic concepts of Majorana theory were introduced by A. A. Ivanov (2009) as a tool to examine the subalgebras of the Griess algebra VM from an elementary axiomatic perspective. A Majorana algebra is a commutative non-associative real algebra generated by a finite set of idempotents, called Majorana axes, that satisfy some properties of Conway’s 2A-axes of VM. If G is a finite group generated by a G-stable set of involutions T, a Majorana representation of (G,T) is an algebra representation of G on a Majorana algebra V together with a compatible bijection between
T and a set of Majorana axes of V . Ivanov’s definitions were inspired by Sakuma’s theorem, which establishes that any two-generated Majorana algebra is isomorphic to one of the Norton-Sakuma algebras. Since then, the construction of Majorana representations of various finite groups has given non-trivial information about the structure of VM.

This thesis concerns two main themes within Majorana theory. The first one is related with the study of some low-dimensional Majorana algebras: the Norton-Sakuma algebras and the Majorana representations of the symmetric group of degree 4 of shapes (2A; 3C) and (2B; 3C). For each one of these algebras, all the idempotents, automorphism groups, and maximal associative subalgebras are described. The second theme is related with a Majorana representation V of the alternating group of degree 12 generated by 11,880 Majorana axes. In particular, the possible linear relations between the 3A-, 4A-, and 5A-axes of V and the Majorana axes of V are explored. Using the known subalgebras and the inner product structure of V , it is proved that neither sets of 3A-axes nor 4A-axes is contained in the linear span of the Majorana axes. When V is a subalgebra of VM, these results, enhanced with information about the characters of the Monster group, establish that the dimension of V lies between 3,960 and 4,689.

P-, T-, C-transformations of the Dirac field in the de Sitter space are studied in the framework of an automorphism set of Clifford algebras. Finite group structure of the discrete transformations is elucidated. It is shown that CP T groups of the Dirac field in Minkowski and de Sitter spaces are isomorphic.

Let $S$ be a class of groups and let $f_S (n)$ be the number of isomorphism classes of groups in $S$ of order $n$. Let $f (n)$ count the number of groups of order $n$ up to isomorphism. The asymptotic bounds for $f (n)$ behave differently when restricted to abelian groups, $A$-groups and groups in general. We survey some results and some open questions in enumeration of finite groups with a focus on enumerating within varieties of $A$-groups.

Universal coverings of the orthogonal groups and their extensions are studied in terms of Clifford-Lipschitz groups. An algebraic description of basic discrete symmetries (space inversion P , time reversal T , charge conjugation C and their combinations P T , CP , CT , CP T) is given. Discrete subgroups {1, P, T, P T } of orthogonal groups of multidimen-sional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. The fundamental automorphisms form a finite group of order 4. The charge conjugation is represented by a complex conjugation pseudoautomor-phism of the Clifford algebra. Such a description allows one to extend the automorphism group. It is shown that an extended automorphism group (CP T-group) forms a finite group of order 8. The group structure and isomorphisms between the extended automorphism groups and finite groups are studied in detail. It is proved that there exist 64 different realizations of CP T-group. An extension of universal coverings (Clifford-Lipschitz groups) of the orthogonal groups is given in terms of CP T-structures which include well-known Shirokov-D¸abrowskiD¸abrowski P T-structures as a particular case. Quotient Clifford-Lipschitz groups and quotient representations are introduced. It is shown that a complete classification of the quotient groups depends on the structure of various subgroups of the extended automorphism group.

Universal coverings of the orthogonal groups and their extensions are studied in terms of Clifford-Lipschitz groups. An algebraic description of basic discrete symmetries (space inversion P , time reversal T , charge conjugation C and their combinations P T , CP , CT , CP T) is given. Discrete subgroups {1, P, T, P T } of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. The fundamental automorphisms form a finite group of order 4. The charge conjugation is represented by a complex conjugation pseudoautomorphism of the Clifford algebra. Such a description allows one to extend the automorphism group. It is shown that an extended automorphism group (CP T-group) forms a finite group of order 8. The group structure and isomorphisms between the extended automorphism groups and finite groups are studied in detail. It is proved that there exist 64 different realizations of CP T-group. An extension of universal coverings (Clifford-Lipschitz groups) of the orthogonal groups is given in terms of CP T-structures which include well-known Shirokov-Dabrowski P Tstructures as a particular case. Quotient Clifford-Lipschitz groups and quotient representations are introduced. It is shown that a complete classification of the quotient groups depends on the structure of various subgroups of the extended automorphism group. An equivalence of the multiplication tables of the groups {1, P, T, P T } and Aut(Cℓ) = {Id, ⋆, , ⋆} proves this isomorphism (in virtue of the commutativity (A ⋆) = (A) ⋆ and the involution conditions (⋆) 2 = () 2

P-, T-, C-transformations of the Dirac field in the de Sitter space are studied in the framework of an automorphism set of Clifford algebras. Finite group structure of the discrete transformations is elucidated. It is shown that CPT groups of the Dirac field in Minkowski and de Sitter spaces are isomorphic.

A group structure of the discrete transformations (parity, time reversal and charge conjugation) for spinor field in de Sitter space are studied in terms of extraspecial finite groups. Two CP T groups are introduced, the first group from an analysis of the de Sitter-Dirac wave equation for spinor field, and the second group from a purely algebraic approach based on the automorphism set of Clifford algebras. It is shown that both groups are isomorphic to each other.

It has been shown by 0. Stephen that the number N of open sets in a non-discrete topology on a finite set with n elements is not greater than 3 x 2. We show that for admissable n/r topologies on a finite group N < 2 , where r is the least order of its non-trivial normal subgroups. This is clearly a sharper bound. Define semi-topological group and topological group as in [2]. (We do not assume that these topological spaces are Hausdorff.) THEOREM 1. Let X be a non-discrete finite semi-topological group of n/r order n (> 1). Then the number N of open seta satisfies N < 2 ' , where r is the least order of the non-trivial normal subgroups of X. Proof. Let 0 be the intersection of all open sets containing x € X. Because there are only a finite number of open sets 0 is open. X We show that 0 (where e is the identity of X) is a normal subgroup. 6