1
$\begingroup$

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.

  • 1
    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?2012-10-20
  • 1
    Sorry, I am a complete math newbie (comp sci). No familiarly with abstract algebras!2012-10-20
  • 0
    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.2012-10-20
  • 2
    I echo my [previous comment](http://math.stackexchange.com/questions/210640/prerequisites-to-category-theory#comment477341_210640) 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.2012-10-20
  • 0
    @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.2012-10-20
  • 0
    That's very good. I should apologize for not including computer science related areas in my list, as it's not something I'm terribly familiar with. (If you are studying CS you might want category theory geared towards your end anyway.)2012-10-20
  • 0
    @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

2 Answers 2

2

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.

  • 0
    Why is it called "permutation"? I still have nightmares about that word from statistics...2012-10-20
  • 1
    @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?2012-10-20
  • 2
    @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.2012-10-20
  • 0
    @RobertMastragostino Thank you, that made it crystal clear!2012-10-20
0

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 $G$.

  • 0
    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