Is a subgroup of a product of two groups necessarily a product of two subgroups ?

Thanks !

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1 Answer

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Not necessarily, Consider the subgroup of Z×Z\mathbb Z\times \mathbb Z generated by (1,1)(1,1)

1

This is the product of Z\mathbb Z with {1}\{1\}.

– Myridium

2 days ago

2

@Myridium what substance are you on?

– Jorge Fernández Hidalgo

2 days ago

Am I making a dumb mistake? The way I interpret the OP’s question is such that in this case, they are looking for a direct product of subgroups of Z\mathbb Z isomorphic to the subgroup of Z×Z\mathbb Z \times \mathbb Z generated by (1,1)(1,1) (which is Z\mathbb Z).

– Myridium

2 days ago

Oh ok, I guess that makes sense. I’ll try to come up with something under that interpretation.

– Jorge Fernández Hidalgo

2 days ago

What about this subgroup of Z² : {(x, x) / x in Z} ?

– Zakaria Oussaad

2 days ago