- #1

- 247

- 0

**Checking to see whether semidirect products are isomorphic.**

## Homework Statement

I want to simplify this semidrect product [itex](Z_7 \rtimes_{\bar{\alpha}} Z_3) \rtimes_{\alpha} Z_2[/itex], but I'm not sure how. In other words, I want to see if this is isomorphic to (for example) [itex]Z_7 \rtimes_{\alpha} Z_6[/itex].

## Homework Equations

## The Attempt at a Solution

I know that [tex]Z_7 \rtimes_{\bar{\alpha}} Z_3[/tex] corresponds to the homomorphism [tex]\bar{\alpha}: Z_3 \rightarrow Z^{\times}_7[/tex], but what homomorphism does [tex](Z_7 \rtimes_{\bar{\alpha}} Z_3) \rtimes_{\alpha} Z_2[/tex] correspond to? I need to know what [tex]Aut(Z_7 \rtimes_{\bar{\alpha}} Z_3)[/tex] is in order to look at [tex]\alpha: Z_2 \rightarrow Aut(Z_7 \rtimes_{\bar{\alpha}} Z_3)[/tex], right? But I'm not sure how to do that...

Thanks in advance

Last edited: