# 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: ${\displaystyle G=ASB}$ for given groups ${\displaystyle A}$ 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 $\displaystyle a,a'\in A$ ;   ${\displaystyle s,s'\in S}$;   ${\displaystyle b,b'\in B}$.

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

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

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

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

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

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

Full text: User:Ian_Clark/IAC72