preview

Example Of Wreath Products : Wreath Product Of Two Finite Groups

Satisfactory Essays
Open Document

Section D: Examples of wreath products

The wreath product of two finite groups:
Definition: Let G and F be two finite groups and suppose that G acts on a finite set X. Denote by F^Xthe set of all maps f:X→F.The set F^Xis a group under pointwise multiplication:(f∙f^ ' )=f(x)f '(x) for all f,f '∈F^Xand x∈X.
We have g(f∙f^ ' )=gf∙gf ' and 〖(gf)〗^(-1)=gf^(-1) in the group; in this way G acts on F^Xas a group of automorphisms.
Define a multiplication on the set F^X×G={(f,g):f∈F^X,g∈G} by setting

(f,g)(f^ ',g^ ' )=(f∙gf^ ',gg^ ') (*)

Then for all (f,g),(f^ ',g^ ')∈F^X×G,where, with the above notation,
(f∙gf^ ' )(x)=f(x)f '(g^(-1) x) for allx∈X. [8]

There are basic properties of the wreath product of finite groups. Suppose the set F^X×G stasfiies (f,g)(f^ ',g^ ' )=(f∙gf^ ',gg^ ') and it forms a group, then the identity element is(1_F,1_G). Where 1_F (x)=1_F for all x∈X, and the inverse of (f,g) is given by (g^(-1) f^(-1),g^(-1)).

Proof: base on the (*) (f,g),(f^ ',g^ ' ),(f^ ' ',g^ ' ' )∈F^X×G
[(f,g)(f^ ',g^ ' )](f^ ' ',g^ ' ') =(f•gf^ ',gg^ ' )(f^ ' ',g^ ' ' ) =[(f•gf^ ' )•gg^ ' f^ ' ',(gg^ ')g^ ' '] ( base on multiplication *) =(f•(gf^ '•gg^ ' f^ ' ' ),(gg^ ')g^ ' ')

Get Access
Related
Get Access