Thread: 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.

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.

Tonio

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?

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 are itself and the trivial subgroup....so what exactly can you conclude from tonio's example?

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?

and this is additive subgroups right?