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

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.

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Full text: User:Ian_Clark/IAC72