- #1

Avatarjoe

- 3

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## Homework Statement

We've shown if G[itex]_{1}[/itex],G[itex]_{2}[/itex],...,G[itex]_{n}[/itex] are subgroups of G s.t.

1)G[itex]_{1}[/itex],G[itex]_{2}[/itex],...,G[itex]_{n}[/itex] are all normal

2)Every element of G can be written as g[itex]_{1}[/itex]g[itex]_{2}[/itex]...g[itex]_{n}[/itex] with g[itex]_{i}[/itex][itex]\in[/itex]G

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

then G[itex]\cong[/itex]G[itex]_{1}[/itex]xG[itex]_{2}[/itex]x...xG[itex]_{n}[/itex]

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

## Homework Equations

## The Attempt at a Solution

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