Homework Help: Application in permutations group

1. Nov 3, 2015

Dassinia

Hello
I am studying for my exam and there's a question that i don't know how to solve, I have some difficulties with symmetric/permutations groups

1. The problem statement, all variables and given/known data

Consider a finite group of order > 2.
We write Aut(G) for the group of automorphisms of G and Sg for the permutations group of the set G.
Consider a1 and a2 ∈ Aut(G) two automorphisms.
We suppose that the g=1 is the only element g ∈ G such as ∀ x ∈ G we have a1(g)*x=x*a2(g)
For g,x ∈ G we have
fg(x):=a1(g)*x*a2(g-1)
Show that for each g ∈ G the application fg: G→G is in Sg

2. Relevant equations

3. The attempt at a solution
I really dont know how to do that..

2. Nov 4, 2015

Staff: Mentor

I don't get the problem here. Since all f_g are well-defined functions on the finite set G with an obvious inverse
h_g (x) := a1(g^-1)*x*a2(g) = f_g^-1 (x), i.e. a bijection, they have to be in Sg, if Sg is Sym(G) the set of all bijections on G.

The additional condition says that f_g = id(G) implies g = e. It is also true that f_g * f_h = f_ (g*h).
Therefore you have a group monomorphism f: G → Sym(G) with f(g) := f_g, i.e. an embedding of G into Sym(G) that respects the group operation.

Last edited: Nov 4, 2015
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