What Are the Non-abelian Groups of a Given Order?

[Jupiter]

Given a group of a certain size, I'm interested in determining whether a non-abelian group of that order exists, and if so how many and what are they up to isomorphism.
For example, for |G|=6, the permutation group S_3 is non-abelian. How can I show that any non-abelian group of order 6 is isomorphic to S_3? Ican show that a given non-abelian group of order six must have 2 elements of order 3, and three of order 2. Now it seems reasonable that I can pair up these elements with the elements of S_3 and show that the two are isomorphic. But, when |G|=8, there are 2 non-abelian groups, D_4 and Q_8. How can I show that any non-abelian, order 8 group is isomorphic to one of these? I imagine as the order increases, the number of possible non-abelian groups will increase, so how is it that I can find them all?
Also, say I'm given a group of order 20. How can I construct/find a non-abelian group of that order?

 

[Janitor]

If there is a systematic way of finding nonabelian groups, I haven't learned about it. That ain't saying much, though.

As far as "How can I construct/find a non-abelian group of ... order [20]?"

Well, there's the dihedral-10 group, which fits the bill. But you knew that already, I'll bet.

 

[matt grime]

Given certain numbers, all groups of that order must be isomorphic. Those are primes and the squares of primes.

For other numbers one needs to consider extensions of groups.

There is always a non-abelian group of order 2n for n>2, the dihedral group.

It suffices to show that for any two distinct odd primes there is a non-abelian extension of one by the other, and also prime powers of order larger than 3. Off the top of my head I don't know that that is true, but it feels true. Try looking up extensions of groups in your favourite algebra source.

You are asking too much to know if there is a way of determining all the structures of possible groups of a given order. It is an open question for p-groups.

 

[Jupiter]

Ok, so I guess I know it's too much to ask, but I just learned how to classify all finite abelian groups, so I figured the next step was to ask what we can know about nonabelian groups. Concerning matt grime's comment that any two groups of order p^2 are isomorphic, I was under the impression that this is not the case. For instance, Z_p^2 and Z_p x Z_p are both groups of order p^2 but they are not isomorphic. Maybe you meant to say "abelian" rather than "isomorphic." But in any event, is this the most we can say about non-abelian groups - that at least one exists for each group of order 2n (n>2)? My motivation here was to try to use what I've learned in group theory to classify small groups (say <30 elements). This shouldn't be too hard since groups of order 1,2,3,4,5,7,9,11,13,15,17,19,23,25 and 29 are all abelian. So all that remains is 6,8,10,12,14,16,18,20,21,22,24,26,27,28.
I looked up "extension" in my text but it came under Chapter 10, "Field Theory." We're just starting ring theory today, so I'll wait on that.

 

[matt grime]

apologies, I meant abelian, not isomorphic.

and the extensions you want arent' field extensions.

classifying possible structures of a given order is computationally very complex (but doable).

also see direct and semi-direct product - classifying certain kinds of extension is as hard as working out automorphism groups and homomorphisms into automorphism groups