Wreath products

The "wreath product" of two groups G and H is built as follows. Express H as a permutation group, permuting n items. Make n copies of G. Let H act on these n copies of G, permuting them. Thus the wreath product of G by H is the semidirect product of (the direct product of n copies of G) by H.

Of course, the above procedure is undefined until we have specified how to express H as a permutation group. The dumb way to do this is to regard H as permuting the elements of H itself (the permutations being given by the Cayley table of H). This is know generally as the "regular wreath product" of G by H, though it could better be called the "dumb wreath product".

Another way is to express H as a permutation group of a small number of items, in as pretty a way as possible. This is still, in general, undefined. For instance, C6 can be expressed as (a b c d e f), or as (a b),(p q r): the former is prettier, but the latter uses fewer items. In general, as here, expressing a group as a transitive set of permutations is prettier than expressing it as an intransitive set.

In the special cases in which H is a symmetric group, "in as pretty a way as possible" is well defined: the symmetric group Sn should be expressed as permutations of n items. The cases where H is the symmetric group Sn and is expressed as permutations of n items give what is known as "permutation wreath products".

Sylow-2-subgroups of Symmetric groups

If H is the prime cyclic group Cp, there is only one sensible way to express it as a permutation group: as one permuting p items. This can be used to build a sequence of groups

Cp
Cp rwp Cp
(Cp rwp Cp ) rwp Cp
((Cp rwp Cp ) rwp Cp) rwp Cp
etc.

which are of interest, as the Sylow-p-subgroup of Spⁿ must be of this form.

This can be proved as follows. Consider the group Tpⁿ, which permutes p sets each of p^n-1 items: the permutations may move items around within their sets, and may interchange sets, but may not lead to mixed sets. Thus Tpⁿ is Sp^n-1 rwp Cp. Recursion then shows that the Sylow-p-subgroup of Tpⁿ must be (...((Cp rwp Cp ) rwp Cp) ... ) rwp Cp. As this is a subgroup of Tpⁿ which is a subgroup of Spⁿ, it must be a subgroup of Spⁿ; and it is big enough (its order is p^2n-1) that it must be the Sylow-p-subgroup of Spⁿ.

An observation

Consider the group Q8. It has eight elements, therefore it can be expressed as permutations of eight items, therefore it is a subgroup of S8.

The Sylow-2-subgroup of S8 is (C2 rwp C2) rwp C2, which is D8 rwp C2, or (D8×D8)⋊C2 with the C2 interchanging the two D8s. Therefore Q8 must be a subgroup of this (D8×D8)⋊C2.

More miscellaneous short pages on finite groups
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