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

• Jan 15th 2010, 07:40 AM
Boysilver
Is there a subgroup H of A4 such that A4 / H isomorphic to S3?
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"?
• Jan 15th 2010, 09:14 AM
tonio
Quote:

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
• Jan 15th 2010, 09:22 AM
Boysilver
Great, thanks.