# Math Help - Fun with normal subgroups.

1. ## Fun with normal subgroups.

This stuff drives me bonkers. Anyway, I have been given or have proved the following:

$\phi : X \rightarrow Y$ is onto.

$x, \bar{x} \in X$

$U \subset X$ is a subgroup

$\phi (U)$ is a subgroup of $Y$

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My last question (for this section) is:

Verify that if $U$ is normal in $X$, then $\phi (U)$ is normal in $Y$

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Can I say this:

If $U$ is normal in $X$ then:

$uXu^{-1}=X$ for all $u \in U$ , so:

$\phi(uXu^{-1})= \phi (X)$

$\phi(u)\phi(X) \phi(u^{-1})= \phi (X)$

$\phi(u) Y \phi(u)^{-1}= Y$ for all $\phi(u) \in \phi(U)$

Hence, $\phi(U)$ is normal in $Y$

2. That's perfect. Of course, it all comes down to the fact that $\phi(u)$ can be any element of $Y$. You should perhaps write that down explicitly.