I know that given a group G and a set S, a group action of G on the S is a function F:S ---> S satisfying some properties.

But what really confuses me arises from a question:

Suppose G is a group and A is an abelian normal subgroup of G, how does G/A operates on A by conjugation?

First, "operates" means "acts"?

If so, then given an element gA of G/A, is the action like... gA$\displaystyle .$a=(gA)a(gA)^{-1}=(gA)a(g^{-1}A)=gAaAg^{-1}=gAg^{-1}=A ?

See, this is where it confuses me.

Group actions by conjugation should take its element gA instead of the basic element ga, but what it follows is that I got a set A as the result of the action instead of an element of A.

I did it by its basic definition, but ended up with a set? Can anyone tell me what's wrong with my equation?