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Classification of groups

  1. Apr 29, 2013 #1
    Checking to see whether semidirect products are isomorphic.

    1. The problem statement, all variables and given/known data

    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].

    2. Relevant equations

    3. 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: Apr 29, 2013
  2. jcsd
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