Counterexample for direct product isomorphism with pairwise trivial intersections

[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}$$\in$G
3)For 1$\leq$i$\leq$n, G$_{i}$$\cap$G$_{1}$,G$_{2}$,...,G$_{i-1}$=e

then G$\cong$G$_{1}$xG$_{2}$x...xG$_{n}$

Show, by example, that if we replace 3) with the condition G$_{i}$$\cap$G$_{k}$=e for all i$\neq$k then G does not need to be isomorphic to G$_{1}$xG$_{2}$x...xG$_{n}$

Homework Equations

The Attempt at a Solution

I tried to find an example with abelian groups like$Z$$_{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??