Revision of A way of getting proper normal subgroups of small index from Sat, 18/04/2009 - 09:38

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Quick description

Given a group, it can be useful to know that there is a proper normal subgroup of small index, where 'small' is defined in terms of some finite invariant of the group. To obtain such a result, it suffices to show that the group has a non-trivial action on a sufficiently small finite set. These may arise from the internal structure of the group.

Prerequisites

Basic group theory

Example 1

This section is incomplete. There need to be specific examples here of applications of this kind of lemma.

Let $G$ be a group with normal (characteristic) subgroups $A$ and $B$ , such that $A$ contains $B$ and $B|=n$ , and such that $A/B$ is a non-central section of G. Then G has a proper normal subgroup of index dividing $(n-1)!$ .

Proof: Since $A/B$ is a non-central section, there is a $G$ -conjugacy class (or $\mathrm{Aut}(G)$ -conjugacy class in the characteristic case) of $A/B$ of order less than $n$ on which $G$ acts non-trivially. This gives a non-trivial ( $\mathrm{Aut}(G)$ -invariant) homomorphism from $G$ to $S_k$ where $1 < k < n$ , and the kernel of this homomorphism is therefore a proper normal (characteristic) subgroup of $G$ of index at most $(n-1)!$ .

For an example of how to use this, suppose $A$ is a group with a characteristic subgroup $B$ of finite index that does not contain the derived subgroup of $A$ . Now suppose $G$ is any group containing $A$ as a normal (characteristic) subgroup, but of arbitrary index. Then the argument above applies, and we obtain a proper normal (characteristic) subgroup $N$ of $G$ of finite index.

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Comments

This looks nice, but I for

Fri, 17/04/2009 - 23:47 — gowers

This looks nice, but I for one will not understand it without some help with basic definitions. Here are the ones I don't know: non-central section (or any kind of section for that matter), derived subgroup. I sort of half remember what a characteristic subgroup is but wouldn't mind being reminded.

Incidentally, when this article is in a more finished state, it will also be a useful contribution to the page on how to use group actions.

Normal subgroups

Sat, 18/04/2009 - 00:20 — tao

Here's a simple result in the same spirit: if H is a subgroup of G of index n, then there is a normal subgroup K of G contained in H of index dividing n!. Proof: G acts by left multiplication on the coset space G/H, which has cardinality n. There are only n! possible ways any group element can act here, so the set of elements which act trivially is a subgroup of index dividing n!. But one easily checks that this subgroup is a normal subgroup that is contained in H.

Normal subgroups

Sat, 18/04/2009 - 22:04 — wilton

For me, it seems unnatural to say this without mentioning the symmetric group. I would give the proof as follows. The action of $G$ on $G/H$ is the same thing as a homomorphism $G\to\mathrm{Sym}(G/H)\cong S_n$ . If $K$ is the kernel of this homomorphism then $G/K$ is isomorphic to a subgroup of $S_n$ and so of order dividing $n!$ . It's clear that $K$ is contained in $H$ (indeed, it's precisely the intersection of $H$ with its conjugates).

I've gone ahead and written

Sat, 18/04/2009 - 23:20 — gowers

I've gone ahead and written up this example as a way of trying to get started an article on proving results by letting a group act on a finite set. I think it's a nice one. (I came across it last year when giving supervisions on a first Cambridge course in group theory: the lecturer had set it as a question.)

Some basic definitions

Sat, 18/04/2009 - 15:06 — Ben Fairbairn (not verified)

Tim wrote:

"I for one will not understand it without some help with basic definitions. Here are the ones I don't know: non-central section (or any kind of section for that matter), derived subgroup. I sort of half remember what a characteristic subgroup is but wouldn't mind being reminded."

Let G be a group.

A SECTION of G is simply a homomorphic image of a subgroup of G (or phrased differently if there is a subgroup H

I see that you too have

Sun, 19/04/2009 - 00:12 — gowers

I see that you too have written up the example that Terence Tao suggested. I think this is absolutely fine: the example can now be found in two genuinely distinct ways: one by somebody who wants to find a normal subgroup of small index (who will naturally arrive at this article) and the other by somebody who wants to understand why people are interested in group actions and how they manage to use them to solve problems that are not ostensibly about actions (who will be led to the article on proving results by letting a group act on a finite set. In general, I am all in favour of the same example popping up in more than one place, and wouldn't even be unduly disturbed if it was simply cut and pasted from one article to another, though that has not happened in this case.

I've rewritten the article in

Sun, 19/04/2009 - 00:14 — Colin Reid

I've rewritten the article in a way that hopefully assumes less background of the reader, and also included Terence's example. This is indeed a very good example of what I am talking about. Is it a bad thing to have it written up here and in Tim's article, or should there just be a link to the most detailed instance of an example (Tim's in this case)?