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## Homework Statement

An isomorphism of a group onto itself is called an automorphism. Prove that the set of all automorphisms of a group is itself a group with respect to composition.

## Homework Equations

To prove that this is a group I must show that it is closed on composition, there is an identity, and each element has an inverse, but proving something is a group isn't where the trouble lies. The trouble lies in reading the problem/understanding the terms.

## The Attempt at a Solution

First let's consider the thing called "automorphism". Is this a mapping? Say, the identity mapping? What exactly is the thing called "automorphism"?

Second, what is the set of all automorphisms of a group? How many ways can you really list the group? Isn't there only one? I'm pretty confused about these meanings. I don't actually need help showing this is a group, I need help knowing what set looks like.