
Normal Subgroups
I have to find the normal subgroups of D_{4}xZ_{2 }and I'm not too sure how to find them. I know that to be a normal subgroup the element has to commute with everything. So I know that I would have to check that (p_{1}, 1) commutes with everything, but I don't know how to check two elements like that at once.
Any hints would be much appreciated.

Re: Normal Subgroups
no, to be a normal subgroup the right and left cosets have to be equal. obviously subgroups that lie in the center of D_{4}xZ_{2} are normal, but there may be others.
some things that may help you:
any subgroup of order 8 will be normal.
any subgroup that is in the center will be normal.
the real trouble is going to be finding the subgroups of order 4 and 2. determining if a subgroup of order 2 is normal will be easy: <a> where a is an element of order 2 is normal if and only if <a> is in the center.