Isomorphism between G and Z x Z_2 if G has a normal subgroup isomorphic to Z_2

[3029298]

Homework Statement

If G contains a normal subgroup H which is isomorphic to \mathbb{Z}_2, and if the corresponding quotient group is infinite cyclic, prove that G is isomorphic to \mathbb{Z}\times\mathbb{Z}_2

The Attempt at a Solution

G/H is infinite cyclic, this means that any g\{h1,h2\} is generated by some \gamma\{h1,h2\} with \gamma\in G. \gamma=g^n because H is normal. But now?

 

[ystael]

Do you know any general property of a group G such that, when G has this property and N is a normal subgroup of G, you can conclude that G \cong N \times G/N?

(Hint: in the direct product H \times K of two groups, what is the relationship between the subgroups H \times 1 and 1 \times K?)

 

[3029298]

Do you know any general property of a group G such that, when G has this property and N is a normal subgroup of G, you can conclude that G \cong N \times G/N?

(Hint: in the direct product H \times K of two groups, what is the relationship between the subgroups H \times 1 and 1 \times K?)

I do not know any general property of this kind... the subgroups H x 1 and 1 x K only have the identity in common and (H x 1)(1 x K)=H x K, but I do not see how this helps...

 

[ystael]

The "general property of G" I was referring to is "G is abelian". One way to understand the thing that makes direct products special is that the factors commute with each other: in the product above, (h, 1)(1, k) = (1, k)(h, 1) = (h, k).

In your original problem, what happens if G is abelian? What happens if it's not?

 

[3029298]

I really do not understand... what is the use of the fact that the factors of the direct products commute with each other?