can G x H1 be isomorphic to G x H2 when H1 not isomorphic to H2?

fromage · January 10th, 2016, 04:54 AM

is the statement:

Suppose H1 is not isomorphic to H2, then GxH1 is not isomorphic to GxH2 for any group G.

True?

This question came from a problem sheet asking if any abelian groups of order 32 are isomorphic-the problem reduced to cases such as showing Z8xZ4 is not isomorphic to Z8xZ2xZ2, which is trivially easy but I was thinking can it be generalised?
Since we didn't get this result in lectures I'm thinking it is not true and I guess a counterexample would be the easiest way to show this but I cant think of any that immediately come to mind so I am asking here.

edit: also what about G1xH1 and G2xH2 where G1 is isomorphic to G2 (same H1 and H2 as above).

Peter · January 10th, 2016, 09:58 AM

Well, the assertion is true if all groups involved are finite abelian groups. Then the fact that H1 and H2 are isomorphic follows from the classification of finite abelian groups.

If we drop the finiteness condition, we can construct a counterexample taking H1 to be any finite Group, H2= H1 x H1 and G to be a countably infinite product of copies of H1.

I guess the assertion is always true as long as G is finite. But I don't see a proof right now.

January 10th, 2016, 04:54 AM	#1
fromage Member Joined: Oct 2013 Posts: 36 Thanks: 0	can G x H1 be isomorphic to G x H2 when H1 not isomorphic to H2? is the statement: Suppose H1 is not isomorphic to H2, then GxH1 is not isomorphic to GxH2 for any group G. True? This question came from a problem sheet asking if any abelian groups of order 32 are isomorphic-the problem reduced to cases such as showing Z8xZ4 is not isomorphic to Z8xZ2xZ2, which is trivially easy but I was thinking can it be generalised? Since we didn't get this result in lectures I'm thinking it is not true and I guess a counterexample would be the easiest way to show this but I cant think of any that immediately come to mind so I am asking here. edit: also what about G1xH1 and G2xH2 where G1 is isomorphic to G2 (same H1 and H2 as above). Last edited by fromage; January 10th, 2016 at 05:21 AM.
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January 10th, 2016, 09:58 AM	#2
Peter Senior Member Joined: Sep 2008 Posts: 150 Thanks: 5	Well, the assertion is true if all groups involved are finite abelian groups. Then the fact that H1 and H2 are isomorphic follows from the classification of finite abelian groups. If we drop the finiteness condition, we can construct a counterexample taking H1 to be any finite Group, H2= H1 x H1 and G to be a countably infinite product of copies of H1. I guess the assertion is always true as long as G is finite. But I don't see a proof right now. Last edited by skipjack; January 10th, 2016 at 02:38 PM.
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