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The group $S_3 \oplus Z_2$ is isomorphic to which of the following groups

  1. $Z_{12}$
  2. $Z_6\oplus Z_2$
  3. $A_4$
  4. $D_6$

Where $\oplus$ is the direct sum. Plz help

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what have you tried? –  Praphulla Koushik Dec 16 '13 at 10:32
Can you rule out any of them for example? –  Tobias Kildetoft Dec 16 '13 at 10:32
I think 1 and 2 are not correct –  EuReka Dec 16 '13 at 10:33
just for clarification... by $D_6$ you mean dihedral group of $6$ elements or of $12$ elements.. ? –  Praphulla Koushik Dec 16 '13 at 10:34
you have two notions... One write $D_n$ to mean dihedral group of order $2n$ and other write $D_{2n}$ to mean dihedral group of order $2n$... I am not sure which one you are using –  Praphulla Koushik Dec 16 '13 at 10:53

1 Answer 1

up vote 3 down vote accepted

This group, $G=S_3\times\mathbb Z_2$ couldn't be isomorphic to $A_4$. Indeed, $A_4$ has only one proper normal subgroups which is $V=\mathbb Z_2\times\mathbb Z_2$, but the group $S_3\times\mathbb Z_2$ has some normal proper subgroups. For example: $\{e\}\times\mathbb Z_2$ or $S_3\times\{1\}$. Also see this good post done by missed Arturo Magidin.

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Nice work, and great link! +1 –  amWhy Dec 16 '13 at 14:24

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