# The Construction of Finite Factorisable Groups

By Ian A. Clark.

An expanded version of a thesis accepted for Ph.D. by the University of Keele, November, 1972.

## ABSTRACT

A theory of non-normal group extension is developed to construct all groups: $G=ASB$ for given groups $A$ and $\displaystyle B$ , and a set of indeterminates $\displaystyle S$ , in which each $\displaystyle g\in G$ has a unique expression as $\displaystyle asb$ , or as $\displaystyle b's'a'$ , where $\displaystyle a,a'\in A$ ;   $s,s'\in S$ ;   $b,b'\in B$ .

A given group $G$ will have this property if and only if it contains $A$ and $B$ as fully-inconjugate subgroups, i.e. for which no non-trivial subgroups of $A$ and $B$ are conjugate in $G$ .

$G$ may be expressed by a generalised transcoset $\{A,S,B,\chi \}$ where $\chi$ is a permutation on the formal product $ASB$ . Under certain conditions the generalised transcoset contains complete information about the structure of $G$ .

The purpose of the present book is to introduce the generalised transcoset as a technique of non-normal group extension for the study of finite simple groups and to investigate its basic properties.

For a given ordered quadruplet $\{A,S,B,\chi \}$ to be a generalised transcoset, $\chi$ must be chosen so that two conditions (the $A$ -commuting condition and the $B$ -commuting condition) are satisfied. The generalised transcoset must also satisfy certain straightforward conditions in order to define a group over $ASB$ . Thus the problem of constructing all groups $G=ASB$ is a special case of the problem of constructing all generalised transcosets $\{A,S,B,\chi \}$ , of interest in its own right.

A transcoset of $A$ is an ordered triplet, $\{A,R,\tau \}$ , satisfying the $A$ -commuting condition. The quadruplet $\{A,S,B,\chi \}$ can be interpreted as simultaneously a transcoset of $A$ , viz $\{A,(SB),\chi \}$ , and a transcoset of $B$ , viz $\{B,(SA),\chi ^{-1}\}$ . All transcosets of $A$ can be partitioned into, and concatenated from, elementary transcosets of $A$ , for which the structure is determined.

The problem is considered of constructing all generalised transcosets of $A$ and $B$ (i.e. transcosets of $A$ which are also transcosets of $B$ ), both for the special case of $S=E$ , which yields $G$ as a Zappa product: $AB|A\cap B=E$ , and for the case of non-trivial $S$ , but with $A$ and $B$ both of order 2, for which a schematic method is introduced.

A computer investigation of the elementary transcosets of $A=D_{4}$ , the dihedral group of order 8, is discussed in the context of Zappa products.

Full text: User:Ian_Clark/IAC72