User:Ian Clark/IAC72

The Construction of Finite Factorisable Groups

By Ian A. Clark.

Abstract.

CHAPTER I. On groups generated by two fully inconjugate subgroups

1.1 Theories of group extension

1.1.1.1. DEFINITION.

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

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

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

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

If use is made of a second group, 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} , in constructing 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} then 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} may be loosely called an extension, or more precisely a group extension, of 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} by 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 in particular cases, a product of 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} 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} .

It is instructive briefly to review previous work on the topic of group extensions of 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} by 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} , 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} 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} are given finite groups.

O. Schreier (1926) showed how to construct all extensions, 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} , of 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} by 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} , 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} becomes a normal subgroup of 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} , 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} becomes the factor group 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/A} . Such extensions are called Schreier extensions.

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

Neither 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} nor 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} need be normal in $G$.

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

1.1.1.2. DEFINITION.

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

Subgroups $A$ and $B$ are called factors of $G$.

1.1.1.3. PROPOSITION.

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

PROOF.

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

$b'=g^{-1}a'g$   for some $g\in G$.

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

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

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

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

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

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, $A$ and $A$. The two sets of permutations are representations of $B$ on the elements of $A$, and representations of $A$ on the elements of $B$, 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, $A\cap B=E$, in order to derive non-simplicity criteria for certain classes of permutable products, $G=AB$.

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

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

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

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

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

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

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

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

by the use of Schreier extension theory. If $G$ is soluble, a fully refined composition series will yield cyclic prime order factor groups, $G_{i-1}/G_{i}$, 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. $Alt_{n}$, provided $n/2$ 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, $p$, dividing $|G|$, there exists a $p$-complement (itself soluble), i.e. a subgroup whose index in $G$ is the highest power of dividing $p$. 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: $PSL_{2,13}$, the projective special linear group of order $(p/2)(p+1)(p-1)$ where $p=13$, is not factorisable. See W.R. Scott (1964). But, taking the same example, $PSL_{2,13}$ may be expressed as $ASB$, where $S$ is a complex of 12 elements, and $A$, $B$ are cyclic subgroups of order 13 and 7 respectively.

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

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 $A\cap B=E$ where $A$ and $B$ are the factors, but so as to express every element of the generalised extension, $G$, of $A$ by $B$ uniquely as either $asb$ or as $b's'a'$, where $S$ is some particular choice of complex in $G$, and $a,a'\in A;s,s'\in S;b,b'\in B$. For such an $S$ to exist it is necessary and sufficient for $A$ and $B$ to be fully inconjugate in $G$, as shown above in subsection 1.2.1.

Our generalised extension theory can construct all finite groups generated by two fully inconjugate subgroups, $A$ and $B$, where $A$ and $B$ 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 $a$ and $b$) depends on whether the cyclic subgroups $A$ and $B$ generated by $a$ and $b$ 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 $A$ by $B$ 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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