How Do Group Homomorphisms Affect Permutation Groups and Subgroups?

[millwallcrazy]

Help with permutation groups...

How do i show that if P: G1 --> G2 is a group homomorphism, then the image, P(G1) =
{g belongs to G2 , s.t. there exists h belonging to G1 , P(h) = g}, is a subgroup of G2

Also if we let G be a group, and Perm(G) be the permutation group of G. How do i show that the map

Q: G --> Perm(G) g --> Qg (g is a subscript of the map Q)
such that Qg(h) = gh is well-defined, 1-1 and a group homomorphism, where
g, h belong to G (again, For Qg(h), the g is a subscript)

Now suppose that G = Z3 (Z subscript 3) = {e, a, a^2}, a^3 = e. If we Label the points of Z3 as {1, 2, 3}, with e = 1, a = 2 and a^2 = 3, how to we give the permutations Qa and Qa^2 , explicitly. (where again a and a^2 are subscripts of Q)

 

[jbunniii]

For the first part, assume you have two elements a, b \in P(G_1).

Thus a = P(x) and b = P(y) for some x,y \in G_1.

What condition(s) must these elements satisfy in order for P(G_1) to be a subgroup of G_2?