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*Algebra*: "Prove that the set ##\operatorname{Aut}(G)## of automorphisms of a group ##G## forms a group, the law of composition being composition of functions."

Of course, we could go through and prove that the four group axioms in the standard definition of a group hold for ##\operatorname{Aut}(G)##. However, I'm lazy and like to do things a little more elegantly.

Suppose we define a group as a category with one object such that all its morphisms are automorphisms. Doesn't the proof follow trivially from this (

*id est*, taking the set of the group as the only object in a category...)?

This is a good lead-in to another, related question: how often is category theory likely to simplify algebraic proofs like this? I was only recently introduced to category theory, and I'm going back through algebra to see if it simplifies things considerably. In other words, how often do situations like the above happen?