Composition Series for Direct Products

[Cryptography]

Homework Statement

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.

 

[mighty2000]

Your attempt at a solution is almost correct. However, there are a few things to consider when finding composition series for a direct product.

Firstly, remember that for a direct product, the order of the factors does not matter. So, for example, Z15xZ6 is the same as Z6xZ15. Therefore, your composition series should also include the following:

5) Z6xZ15 > Z2xZ5 > {0,0}
6) Z6xZ15 > Z3xZ5 > {0,0}

Secondly, when finding composition series for a direct product, you need to consider all possible combinations of composition series for the individual groups. In your attempt, you have only considered one composition series for Z15 and one for Z6, but there are actually more possibilities.

So, the complete list of composition series for Z15xZ6 would be:

1) Z15xZ6 > Z5xZ3 > {0,0}
2) Z15xZ6 > Z5xZ2 > {0,0}
3) Z15xZ6 > Z3xZ3 > {0,0}
4) Z15xZ6 > Z3xZ2 > {0,0}
5) Z6xZ15 > Z2xZ5 > {0,0}
6) Z6xZ15 > Z3xZ5 > {0,0}

And the corresponding composition factors would be:

1) Z3xZ2, Z5xZ3
2) Z3xZ3, Z5xZ2
3) Z5xZ2, Z3xZ3
4) Z5xZ3, Z3xZ2
5) Z2xZ3, Z5xZ5
6) Z3xZ2, Z5xZ5

I hope this helps clarify things for you. Remember, when finding composition series for a direct product, make sure to consider all possible combinations and also keep in mind that the order of the factors does not matter. Good luck with your studies!