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Question: Give an example of a normal subgroup of $A_4$. Is the subgroup
$H$={$(1),(25)(34),(35)(24),(45)(23)$} of $A_5$ a normal subgroup?

I am having a hard time understanding this concept because the question itself is confusing me. Do I have to come up with a group showing $A_4$? Is $A_4$=[$(1),(12)(34),(13)(24),(123)$}? Need help

Lady T
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  • $A_5$ is a simple group, so it has no proper non-trivial normal subgroups. You're looking for subgroups of $A_4$ anyway! – Edward Evans Apr 17 '17 at 16:10
  • How do I go about finding that out? For example: Can $A_4$={$(1),(12)(34),(13)(24),(123)$}? – Lady T Apr 17 '17 at 16:15
  • Take the Klein four group $K = { (1), (12)(34), (13)(24), (14)(23)}$. – Dietrich Burde Apr 17 '17 at 16:24
  • After that, I can just say $H$ is not in normal in $A_4$? – Lady T Apr 17 '17 at 16:26
  • @LadyT Are you familiar with the concept of a normal subgroup? A subgroup $H$ of a group $G$ is called normal if for any $g \in G$, the left and right cosets of $H$ coincide, that is

    $$gH = Hg.$$

    We then write $H \trianglelefteq G$.

     – Edward Evans Apr 17 '17 at 16:30
  • @ÍgjøgnumMeg know the definition but incorporating into a problem is give me trouble – Lady T Apr 17 '17 at 16:35
  • @ÍgjøgnumMeg I see now, I have to take the conjugation of an element in $H$ to see if $H$ is $A_5$ – Lady T Apr 17 '17 at 17:23

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