# Group and Direct Products

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• December 16th 2009, 05:42 PM
pleasehelpme1
Group and Direct Products
Let G be a group and let G=HxK. Give an example where J is a subgroup of G, but not of the form H_1xK_1 where H_1 is a subgroup of H and K_1 is a subgroup of K.
• December 16th 2009, 06:21 PM
tonio
Quote:

Originally Posted by pleasehelpme1
Let G be a group and let G=HxK. Give an example where J is a subgroup of G, but not of the form H_1xK_1 where H_1 is a subgroup of H and K_1 is a subgroup of K.

$\{(0,0)\,,\,(1,1)\} \le \mathbb{Z}_2\times \mathbb{Z}_2$

Tonio
• December 16th 2009, 06:47 PM
pleasehelpme1
???
what exactly do you mean by this? can you prove this for me? and what are the subgroup H_1 and K_1 in this case?
• December 16th 2009, 07:52 PM
Drexel28
Quote:

Originally Posted by pleasehelpme1
what exactly do you mean by this? can you prove this for me? and what are the subgroup H_1 and K_1 in this case?

The only subgroups of $\mathbb{Z}_2$ are itself and the trivial subgroup....so what exactly can you conclude from tonio's example?
• December 17th 2009, 04:53 AM
pleasehelpme1
...
That H_1 and K_1 must either be Z_2 or {0}. However, then J=Z_2xZ_2 or {0}x{0} contradicting the definition of J?
• December 17th 2009, 07:03 AM
pleasehelpme1
..
and this is additive subgroups right?