(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Write down all composition series for the group Z_{15}xZ_{6}(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 Z_{15}:

Z_{15}> Z_{5}> {0}

and

Z_{15}> Z_{3}> {0}

and I understand that the composition factors are the possible factor groups i.e. Z_{3}and Z_{5}

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 Z_{15}xZ_{6}are:

1) Z_{15}xZ_{6}> Z_{5}xZ_{3}> {0,0}

2) Z_{15}xZ_{6}> Z_{5}xZ_{2}> {0,0}

3) Z_{15}xZ_{6}> Z_{3}xZ_{3}> {0,0}

4) Z_{15}xZ_{6}> Z_{3}xZ_{2}> {0,0}

and that the composition factors for each one respectively are:

1) Z_{3}xZ_{2}, Z_{5}xZ_{3}

2) Z_{3}xZ_{3}, Z_{5}xZ_{2}

3) Z_{5}xZ_{2}, Z_{3}xZ_{3}

4) Z_{5}xZ_{3}, Z_{3}xZ_{2}?

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

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# Homework Help: Composition Series for Direct Products

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