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EDIT: I am a complete math newbie with only calculus and linear algebra under my belt. One of the reasons I chose that book - as it's aimed at general audience.

Reading Awodey's Category Theory he says, on p. 12 (NOTE: I provide the definition for the purposes of the context only. I have no questions about that):

Definition 1.4. A group $G$ is a monoid with an inverse $g^{-1}$ for every element $g$. Thus, $G$ is a category with one object, in which every arrow is an isomorphism.

For any set $X$, we have a group $\operatorname{Aut}(X)$ of automorphisms (or "permutations") of $X$, that is, isomorphisms $f:X\to X$. A group of permutations is a subgroup $G\subseteq \operatorname{Aut}(X)$ for some set $X$, that is, a group of (some) automorphisms of $X$. Thus, the set $G$ must satisfy the following...

  1. What are these "permutations/automorphisms"? He never introduced them before.
  2. And what is a "subgroup"? Is it used informally here?
  3. I guess more general question, is what is the author trying to teach me here - the fact the group can have subgroups...?

Any help appreciated, thanks.

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    @anon No need, I appreciate any advice or pointers from people much smarter than me :-) On your last point, unfortunately there are no good materials teaching category theory from a comp. sci. point of view, mainly because it only tends to confuse. Also, most of the "simplest" things in a language like Haskell (e.g. lists) tend to require rather advanced concepts of category theory (e.g. free monoids, but finite). However, I would agree with you 100% that Category Theory on it's own would make absolutely no sense whatsoever!2012-10-20

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He introduced the permutations in that definition: a permutation of $X$ is a bijection $X\to X$. A subgroup $H$ of $G$ is just a subset $H\subset G$ that is also a group with the same operations.

What the authors might be trying to teach you is that there are a lot of groups. You will eventually learn, also, that any finite group is a subgroup of some group of permutations like those in that definition.

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    @RobertMastragostino Thank you, that made it crystal clear!2012-10-20
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An automorphism is a bijective map from a group $G$ to itself that preserves the operation. A subgroup $H$ of a group $G$ is a subset of $G$ such that $H$ is itself a group under the operation of $G$.

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    You're talking about group automorphisms, while the OP's text is talking about concrete groups as being comprised of *set* automorphisms.2012-10-20