## Representation Theory of Semidirect Products: The Preliminaries (Pt. II)

**Point of Post: **This post is a continuation of this one.

What we next do is ‘dualize’ the action to an action (where we, for notational convenience, have denoted by . Since is abelian we know the two coincide) by . It is not at all clear that 1) is really a homomorphism, 2) is an automorphism, and 3) is a homomorphism. Indeed:

**Theorem: ***Let be defined as above. Then:*

**Proof:**

**1) **We note that for any , , and one has that

(note the use of the abelianess of ).

**2) **We note that for any , , and any one has

so that is an endomorphism. To prove it is an automorphism it suffices to prove (since is finite) to prove it is injective. To do this we note that if and then and so

Thus, for every , and thus .

**3)** We note that for any and one has

and so .

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

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