# User:Ian Clark/IAC72

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# The Construction of Finite Factorisable Groups

By Ian A. Clark.

## CHAPTER I. On groups generated by two fully inconjugate subgroups

### 1.1 Theories of group extension

1.1.1.1. DEFINITION.

Two subgroups, $\displaystyle A$ and $\displaystyle B$ , of a group $\displaystyle G$ are fully inconjugate in $\displaystyle G$ if and only if no proper subgroup of $\displaystyle A$ is conjugate in ${\displaystyle G}$ to a proper subgroup of ${\displaystyle B}$.

it follows immediately that if ${\displaystyle A}$ and ${\displaystyle B}$ are both finite and of relatively prime order then they are fully inconjugate in ${\displaystyle G}$.

In the present work we will mostly confine our attention to this situation, addressing the problem of constructing all finite groups, ${\displaystyle G}$, which are generated by two given finite nontrivial groups, $\displaystyle A$ and $\displaystyle B$ , such that $\displaystyle A$ and $\displaystyle B$ are fully inconjugate in $\displaystyle G$ .

This problem is a special case of embedding a group $\displaystyle A$ in a group $\displaystyle G$ as a proper subgroup. Then $\displaystyle G$ is sometimes called an extension of $\displaystyle A$ (e.g. by M. Hall, jnr (1959)).

If use is made of a second group, $\displaystyle B$ , in constructing $\displaystyle G$ then $\displaystyle G$ may be loosely called an extension, or more precisely a group extension, of $\displaystyle A$ by $\displaystyle B$ . And in particular cases, a product of $\displaystyle A$ and $\displaystyle B$ .

It is instructive briefly to review previous work on the topic of group extensions of $\displaystyle A$ by $\displaystyle B$ , where $\displaystyle A$ and $\displaystyle B$ are given finite groups.

O. Schreier (1926) showed how to construct all extensions, $\displaystyle G$ , of $\displaystyle A$ by $\displaystyle B$ , where $\displaystyle A$ becomes a normal subgroup of $\displaystyle G$ , and $\displaystyle B$ becomes the factor group $\displaystyle G/A$ . Such extensions are called Schreier extensions.

G. Zappa (1940) treated the class of permutable products of $\displaystyle A$ by $\displaystyle B$ . These are groups, $\displaystyle G$ , in which every element of $\displaystyle G$ can be expressed as $\displaystyle ab$ , or as $\displaystyle b'a'$ , for some $\displaystyle a,a' \in A; b,b' \in B$ . Writing $\displaystyle AB$ for the set $\displaystyle \{ab | a,a' \in A; b,b' \in B\}$ , then $\displaystyle G=AB=BA$ .

Neither $\displaystyle A$ nor $\displaystyle B$ need be normal in ${\displaystyle G}$.

Zappa imposed the restriction ${\displaystyle A\cap B=E}$, the trivial group. Under this restriction, each ${\displaystyle g\in G}$ has a unique expression in either form: ${\displaystyle ab}$ or ${\displaystyle b'a'}$.

1.1.1.2. DEFINITION.

A Zappa product is a group ${\displaystyle G=AB=BA|A\cap B=E}$.

Subgroups ${\displaystyle A}$ and ${\displaystyle B}$ are called factors of ${\displaystyle G}$.

1.1.1.3. PROPOSITION.

If ${\displaystyle G}$ is a Zappa product of factors ${\displaystyle A}$ and ${\displaystyle B}$, then ${\displaystyle A}$ and ${\displaystyle B}$ are fully inconjugate.

PROOF.

Otherwise for some non-trivial ${\displaystyle a'\in A}$; ${\displaystyle b'\in B}$;

${\displaystyle b'=g^{-1}a'g}$   for some ${\displaystyle g\in G}$.

Now ${\displaystyle g=ab}$   for some ${\displaystyle a\in A}$;   ${\displaystyle b\in B}$.

Therefore ${\displaystyle b'=(ab)^{-1}a'(ab)}$   ${\displaystyle =b^{-1}a^{-1}a'ab}$,

whence ${\displaystyle bb'b^{-1}=a^{-1}a'a}$     ${\displaystyle \neq 1}$,

contradicting the assumption of ${\displaystyle A\cap B=E}$.

The Zappa product falls within the scope of the present work as a special case: ${\displaystyle S=E}$.

The matter will be treated further in Chapter 3.

Zappa obtained necessary and sufficient conditions upon two sets of permutations for them to define a Zappa product of the two given finite groups, ${\displaystyle A}$ and ${\displaystyle A}$. The two sets of permutations are representations of ${\displaystyle B}$ on the elements of ${\displaystyle A}$, and representations of ${\displaystyle A}$ on the elements of ${\displaystyle B}$, respectively. Zappa's conditions bear a presumably intentional resemblance to the necessary and sufficient conditions for a given automorphism group and factor set to define a Schreier extension of one given group by another.

J Szép and L Rédei (1950, 1951) generalised the work of Zappa by relaxing the condition, ${\displaystyle A\cap B=E}$, in order to derive non-simplicity criteria for certain classes of permutable products, ${\displaystyle G=AB}$.

L Rédei (1950) introduced a highly generalised form of extension into Group Theory called the skew product.

The skew product of ${\displaystyle A}$ by ${\displaystyle B}$, where ${\displaystyle A}$ and ${\displaystyle B}$ are groups, is defined as the set of ordered couples: ${\displaystyle \{[ab]|a\in A;b\in B\}}$ under a binary operation: ${\displaystyle \circ }$, defined by

${\displaystyle [a_{1},b_{1}]\circ [a_{2},b_{2}]=[\alpha (a_{1},b_{1},a_{2},b_{2}),\beta (a_{1},b_{1},a_{2},b_{2})]}$

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are functions mapping the domain of arguments ${\displaystyle (A,B,A,B)}$ onto ${\displaystyle A}$ and ${\displaystyle B}$ respectively.

Rédei exhibited a particular case of the skew product where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are of a certain restricted form, for which he obtained a necessary and sufficient set of 16 conditions for ${\displaystyle A\circ B}$ to be a group. This restricted skew product is noteworthy because the Zappa product of ${\displaystyle A}$ and ${\displaystyle B}$ and the Schreier extension of ${\displaystyle A}$ by ${\displaystyle B}$ are both special cases of ${\displaystyle A\circ B}$.

Rédei thus succeeds in generalising Schreier extension theory to include the Zappa product, ${\displaystyle AB}$, as a (generalised) extension of ${\displaystyle A}$ by ${\displaystyle B}$.

What might be the value of doing so? Quite apart from the conceptual elegance of a unified theory of group extension, irrespective of how inconvenient it may be in practical use, there is the attraction of finding an analogy to the composition series of a group by using the concept of the Zappa product factor instead of that of the normal subgroup.

A finite group, ${\displaystyle G,}$ may be reassembled from the factor groups, ${\displaystyle G_{i-1}/G_{i}}$, of its composition series:

${\displaystyle G=G_{0}\triangleright G_{1}\triangleright G_{2}\triangleright \cdots \triangleright G_{n}=1}$

by the use of Schreier extension theory. If ${\displaystyle G}$ is soluble, a fully refined composition series will yield cyclic prime order factor groups, ${\displaystyle G_{i-1}/G_{i}}$, otherwise some of these will be simple groups (of composite order).

Simple groups have only trivial composition series. However many simple groups have a non-trivial expression as a repeated Zappa product (e.g. ${\displaystyle Alt_{n}}$, provided ${\displaystyle n/2}$ is not an odd integer). A solvable group of composite order may be expressed as a repeated Zappa product of its co-prime order Sylow subgroups, since for every prime number, ${\displaystyle p}$, dividing ${\displaystyle |G|}$, there exists a ${\displaystyle p}$-complement (itself soluble), i.e. a subgroup whose index in ${\displaystyle G}$ is the highest power of dividing ${\displaystyle p}$. See, for example: M Hall, jnr (1959) p. 41 et seq).

Hence Zappa product extension theory allows not only all solvable groups to be constructed from their Sylow subgroups, but some simple groups as well.

However, not all simple groups are expressible as permutable products, let alone as a product of which one factor is a Sylow subgroup. For example: ${\displaystyle PSL_{2,13}}$, the projective special linear group of order ${\displaystyle (p/2)(p+1)(p-1)}$ where ${\displaystyle p=13}$, is not factorisable. See W.R. Scott (1964). But, taking the same example, ${\displaystyle PSL_{2,13}}$ may be expressed as ${\displaystyle ASB}$, where ${\displaystyle S}$ is a complex of 12 elements, and ${\displaystyle A}$, ${\displaystyle B}$ are cyclic subgroups of order 13 and 7 respectively.

Each element in ${\displaystyle PSL_{2,13}}$ then has a unique expression in the form ${\displaystyle asb=b's'a'}$ for some ${\displaystyle a,a'\in A}$;   ${\displaystyle s,s'\in S}$;   ${\displaystyle b,b'\in B}$, provided ${\displaystyle S}$ is a common transversal of the two sets of distinct double-cosets, ${\displaystyle \{AgB|g\in G\},\{Bg'A|g'\in G\}}$

The need thus arises for an extension theory which can handle simple groups which are not Zappa products. The theory here presented may be viewed as a generalisation of the Zappa product: not in the obvious manner of relaxing the condition ${\displaystyle A\cap B=E}$ where ${\displaystyle A}$ and ${\displaystyle B}$ are the factors, but so as to express every element of the generalised extension, ${\displaystyle G}$, of ${\displaystyle A}$ by ${\displaystyle B}$ uniquely as either ${\displaystyle asb}$ or as ${\displaystyle b's'a'}$, where ${\displaystyle S}$ is some particular choice of complex in ${\displaystyle G}$, and ${\displaystyle a,a'\in A;s,s'\in S;b,b'\in B}$. For such an ${\displaystyle S}$ to exist it is necessary and sufficient for ${\displaystyle A}$ and ${\displaystyle B}$ to be fully inconjugate in ${\displaystyle G}$, as shown above in subsection 1.2.1.

Our generalised extension theory can construct all finite groups generated by two fully inconjugate subgroups, ${\displaystyle A}$ and ${\displaystyle B}$, where ${\displaystyle A}$ and ${\displaystyle B}$ are given. Whether this includes all finite non-cyclic simple groups is still an open question, though the fact (recently proven) that all such groups can be generated by just 2 elements (call them ${\displaystyle a}$ and ${\displaystyle b}$) depends on whether the cyclic subgroups ${\displaystyle A}$ and ${\displaystyle B}$ generated by ${\displaystyle a}$ and ${\displaystyle b}$ respectively can always be chosen to be fully inconjugate.

Another motive for developing the present theory is to facilitate the actual construction of extensions of ${\displaystyle A}$ by ${\displaystyle B}$ having particular properties (e.g. see subsection 2.2.6.). This is something not obviously accessible to either Zappa or Schreier extension theories as they stand.

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