# Examples of non abelian p-group

i'm working on a problem that asks me to find a principal series for a p-group where the subgroups in the series have orders consecutive powers of p.

To help me think about this problem, I would like one or more examples of non abelian p-groups to work with.

The only non abelian groups I've seen are permutation groups, alternating group, and groups of rotations and symmetries of geometric objects like the square, traingle, and cube. None of these are p-groups. Can anyone give an example of a nonabelian p-group?

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Actually the group D_4 of rotations and symmetries of a square is a nonabelian 2-group (it has order 8). The other nonabelian group of order 8 is the quaternion group Q. So here are two examples.

What else is there... A group G of order p^2 is necessarily abelian: Z(G) is nontrivial so G/Z(G) is either trivial or cyclic of prime order, and since the latter case can't happen, we must have that G=Z(G).

So the next candidates will have order 16. The group D_8 of rotations and symmetries of an octagon is one such nonabelian group. We also get two freebies from the order 8 case: take Z_2 x D_4 and Z_2 x Q. There are probably some more.

If you continue this way you can get a good bunch of concrete nonabelian p-groups (especially 2-groups). For a more abstract approach, you can try taking semidirect products of things.

(And for infinite p-groups you'll probably want to approach this differently.)

mathwonk