Constructing Non Abelian Groups of Given order

Question

The motivating factor behind this question is the comments given for this question asked some hours ago Group of order $105$

So, given a group $G$ of order $n$, what are different methods for constructing a non abelian group of order $n$? Well, I have seen a method which Herstein uses in his book. He takes a cyclic group of given order and defines an Automorphism on the cyclic group and then places some restrictions.

I would like to know whether there are anymore methods for obtaining a Non Abelian group of a given order.

See http://mathoverflow.net/questions/31553/finite-nonabelian-groups-of-odd-order/31558#31558 — Robin Chapman, Oct 11 '10 at 18:37
Also http://mathoverflow.net/questions/11001/non-abelian-groups-of-prescribed-order. — Arturo Magidin, Oct 11 '10 at 21:09

score 3 · Accepted Answer · answered Nov 25 '10 at 19:24

3

Have you tried taking the semi-direct product of two subgroups (one needs to be a normal subgroup). The order of G is the product of the orders of the subgroups. One subgroup needs to be normal. If the other is not, then the semidirect product is usually (if not always) non-abelian

answered Nov 25 '10 at 19:24

user3180

2,372

1

Well, yes, but to get a non-Abelian semidirect product, you need to know that there is some non-trivial homomorphism from that group into the automorphism group of the normal one, which is the hard part- there's always a homomorphism with kernel the whole group, which leads to the direct product. – Geoff Robinson Jul 08 '11 at 10:44

Constructing Non Abelian Groups of Given order

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