Is there a subgroup H of A4 such that A4 / H isomorphic to S3?

**Boysilver** · January 15th 2010, 06:40 AM

Is there a subgroup $H$ of $A_4$ such that the quotient group ${A_4}/H \cong S_3$ ? I know I could just go through the possibilities, but is there a very quick reason why the answer should be "no"?

**tonio** · January 15th 2010, 08:14 AM

Originally Posted by Boysilver

Is there a subgroup $H$ of $A_4$ such that the quotient group ${A_4}/H \cong S_3$ ? I know I could just go through the possibilities, but is there a very quick reason why the answer should be "no"?

As quick as possible: $A_4$ has no normal subgroups of order 2.

Tonio

**Boysilver** · January 15th 2010, 08:22 AM

Great, thanks.

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