# User:Ian Clark/IAC72/Abstract

# 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:
for given groups and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}**
,
and a set of indeterminates **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}**
,
in which each **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in G}**
has a unique expression as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle asb}**
, or as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b's'a'}**
, where
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,a'\in A}**
; ; .

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

may be expressed by a *generalised transcoset*
where
is a permutation on the formal product .
Under certain conditions the generalised transcoset contains complete information about the structure of .

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 to be a generalised transcoset, must be chosen so that two conditions (the -commuting condition and the -commuting condition) are satisfied. The generalised transcoset must also satisfy certain straightforward conditions in order to define a group over . Thus the problem of constructing all groups is a special case of the problem of constructing all generalised transcosets , of interest in its own right.

A *transcoset* of is an ordered triplet, , satisfying the -commuting condition.
The quadruplet
can be interpreted as simultaneously a transcoset of , viz
,
and a transcoset of , viz
.
All transcosets of can be partitioned into, and concatenated from, elementary transcosets of , for which the structure is determined.

The problem is considered of constructing all generalised transcosets of and (i.e. transcosets of which are also transcosets of ), both for the special case of , which yields as a Zappa product: , and for the case of non-trivial , but with and both of order 2, for which a schematic method is introduced.

A computer investigation of the elementary transcosets of , the dihedral group of order 8, is discussed in the context of Zappa products.

Full text: User:Ian_Clark/IAC72