# Composition Series for Direct Products

1. Nov 26, 2011

### Cryptography

1. The problem statement, all variables and given/known data

Write down all composition series for the group Z15xZ6 (the direct product of the groups of integers modulo 15 and 6), and then by considering the composition series, give a list of composition factors.

2. The attempt at a solution

I understand how to give composition series for individual groups:

e.g. for Z15:

Z15 > Z5 > {0}
and
Z15 > Z3 > {0}

and I understand that the composition factors are the possible factor groups i.e. Z3 and Z5

But for a direct product I'm a little confused.

I know that if A is a subgroup of G and B is a subgroup of H, then AxB is a subgroup of GxH. So does this mean that the composition series for Z15xZ6 are:

1) Z15xZ6 > Z5xZ3 > {0,0}
2) Z15xZ6 > Z5xZ2 > {0,0}
3) Z15xZ6 > Z3xZ3 > {0,0}
4) Z15xZ6 > Z3xZ2 > {0,0}

and that the composition factors for each one respectively are:

1) Z3xZ2, Z5xZ3
2) Z3xZ3, Z5xZ2
3) Z5xZ2, Z3xZ3
4) Z5xZ3, Z3xZ2 ?

Sorry if I've done something completely wrong and thank you very much for your help.