Group Extensions: A Simple Guide

[calvino]

A group G is an extension of A by B (A,B groups) if there exists a normal subgroup of G (call it N) such that A is isomorphic to N, and G/N is isomorphic to B.

Is there a simple way to identify ALL such extensions, if A, B are small groups (order 4 or less would dignify "small"). Simple or not, how is it done? -I've looked throughout the web, and cannot find much on "group extensions".

Furthermore, is there a way to say how many extensions of A by B exist?

I decided to try a simple example (finding as many non-isomorphic extensions of Z_2 by Z_2 (integers mod 2), and obviously there's the Z_2 X Z_2 extension, and there's Z_4 . What other possibilities are there?

 

[Hurkyl]

Well, if A is isomorphic to N, and B is isomorphic to G/N, then you know:

|A| = |N|
|B| = |G| / |N|
and so |G| = |A| |B|

which ought to help you in your classification efforts.

I decided to try a simple example (finding as many non-isomorphic extensions of Z_2 by Z_2 (integers mod 2), and obviously there's the Z_2 X Z_2 extension, and there's Z_4 . What other possibilities are there?

Since you've just listed every group of order 4, you've covered all the possibilities.


It might be worthwhile to consider the case where G is abelian (and thus so are A and B), because of the nice structure theorems for abelian groups.

I suspect that if you're really good at this stuff, the finite, abelian case could be solved completely over the course of an afternoon!

Anyways, the programme, I expect, would be to build up the full answer by starting with cyclic groups of prime order, then of prime power order, and then showing how the theory behaves when you take the products of groups.

 

[matt grime]

Yes, this theory is completely classified, by group cohomology. The general case is hard to describe, but the cases you mention are easy.