# What are automorphisms and subgroups?

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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It would be helpful to know your background, in part because your tone is difficult to interpret. For example, are you familiar with groups from abstract algebra? – Adam Saltz Oct 20 '12 at 2:10
Sorry, I am a complete math newbie (comp sci). No familiarly with abstract algebras! – drozzy Oct 20 '12 at 2:14
Trying to pick up and read a book on category theory before being familiar with some of the big motivating ideas in algebra and topology seems... counterproductive. – kahen Oct 20 '12 at 2:30
I echo my previous comment about foraging into category theory: it is something I would not recommend until you already understand the "chapter 0-1" basics in a handful of different and varied subjects. This allows you to be familiar with a stockpile of categories already and understand the constructions and thinking involved in category theory, which is invaluable since it is not easy to absorb by the uninitiated. – anon Oct 20 '12 at 2:40
@anon Thank you, however in my case I have plenty of examples and motivation to draw from in my field, through the use of Haskell programming language. – drozzy Oct 20 '12 at 4:16

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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Why is it called "permutation"? I still have nightmares about that word from statistics... – drozzy Oct 20 '12 at 2:23
@drozzy: When you shuffle the order of cards in a deck, or rearrange the furniture in your room by switching their positions, you are permuting things. Are you familiar with the English word "permutation" outside of mathematics? – anon Oct 20 '12 at 2:25
@drozzy a bijection is a function that maps one-to-one between sets, never leaving anything out in either. A bijection $X\to X$ therefore sends things to different places, e.g $\{1,2,3\}\to\{2,3,1\}$ is a bijection on the set $\{1,2,3\}$ because it sends the set to itself in some way. – Robert Mastragostino Oct 20 '12 at 2:42
@RobertMastragostino Thank you, that made it crystal clear! – drozzy Oct 20 '12 at 4:19

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. – anon Oct 20 '12 at 2:21