Abstract Algebra homework Direct products

[Avatarjoe]

Homework Statement

We've shown if G_{1},G_{2},...,G_{n} are subgroups of G s.t.

1)G_{1},G_{2},...,G_{n} are all normal
2)Every element of G can be written as g_{1}g_{2}...g_{n} with g_{i}\inG
3)For 1\leqi\leqn, G_{i}\capG_{1},G_{2},...,G_{i-1}=e

then G\congG_{1}xG_{2}x...xG_{n}

Show, by example, that if we replace 3) with the condition G_{i}\capG_{k}=e for all i\neqk then G does not need to be isomorphic to G_{1}xG_{2}x...xG_{n}

Homework Equations

The Attempt at a Solution

I tried to find an example with abelian groups likeZ_{60}, but nothing seemed to work. Now I'm trying groups that are themselves direct products, but I seem to be missing the big picture.

 

[micromass]

You need to find a good homomorphism

\varphi:G_1\times...\times G_n\rightarrow G

and show that that is an isomorphism. What do you think you can choose as \varphi??

 

[micromass]

You also might want to think about an induction on n. Can you show it for n=2??