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## Main Question or Discussion Point

I have a hard time trying to figure out these types of problems.

Usually, we are supposed to compare their orders, compare if they are both abelian, or see if one group has element of order n which the other group does not have.

So in this case, A4 has order 12 and D3xZ2 also has order 12. So this doesn't help.

An alternating group is non-abelian for n<=3 so A4 is non-abelian. D3 is non-abelian as well and the product of non-abelian to a group is non-abelian (?). So this doesn't help.

So what I'm thinking is, it's either "A4 has an element of order 4, but D3xZ2 does not", or "D3xZ2 has element order 6 but A4 does not".

So my question is, how do we find that D3xZ2 does or does not contain element of order 6? What if it was Z3xZ2 or S4xD3?

Are there any tricks to figuring out if a product of two groups has element of order n so that I can compare it with the other group, in this case A4?

Thank you

Usually, we are supposed to compare their orders, compare if they are both abelian, or see if one group has element of order n which the other group does not have.

So in this case, A4 has order 12 and D3xZ2 also has order 12. So this doesn't help.

An alternating group is non-abelian for n<=3 so A4 is non-abelian. D3 is non-abelian as well and the product of non-abelian to a group is non-abelian (?). So this doesn't help.

So what I'm thinking is, it's either "A4 has an element of order 4, but D3xZ2 does not", or "D3xZ2 has element order 6 but A4 does not".

So my question is, how do we find that D3xZ2 does or does not contain element of order 6? What if it was Z3xZ2 or S4xD3?

Are there any tricks to figuring out if a product of two groups has element of order n so that I can compare it with the other group, in this case A4?

Thank you