Is there a classification of all finite indecomposable p-groups? (something like: those are exactly the quaternions, primary cylic groups, some dihedral groups, etc.).
I'll give some examples of indecomposable p-groups, a survey of small orders, and an asymptotic description. As Derek Holt points out, most p-groups are directly indecomposable, and perhaps he has a cleaner way of describing the third section.
Some indecomposable finite p-groups:
- cyclic groups
- p-groups of maximal class (with nilpotency class n and order pn+1), including:
- dihedral groups (of order at least 8)
- quaternion groups
- semi-dihedral groups
- extra-special groups with a center of order p and elementary abelian quotient
- Any group with a cyclic center
Every group of order p is directly indecomposable (being cyclic).
One out of two groups of order p2 is directly indecomposable (the cyclic one).
Three out of five groups of order p3 are directly indecomposable (the cyclic one, and the two non-abelian ones which are both extra-special and maximal class).
Eight out of fourteen groups of order 16 and nine out of fifteen groups of order p4 for p odd are directly indecomposable (all but two with cyclic center).
34 out of 51 of order 32, 49 out of 67 of order 243, 59 out 77 for of order 3125, and then a steady pattern out of 61 + 2p + 2(3,p−1) + (4,p−1).
While the groups are organized into families, even the number of families becomes unmanageable after a point and so a different technique is needed for larger n.
Where do decomposable p-groups come from? They are direct products of smaller p-groups. However, there are about p(2n3/27) groups of order pn, and so taking direct products of groups of order pi with groups of order pn−i for 1 ≤ i ≤ n/2 gives a somewhat complicated sum but which is (for large n) less than 1/pn th as big as p(2n3/27). In other words, a vanishingly small fraction of p-groups are decomposable, and the rest are indecomposable.