I'm asked to find a group G with a subgroup H such that there is no normal subgroup N of G which performs: G/N =~ H.
I thought of G=S4 and H=A4, because I don't think there is an homomorphism from S4 to A4, but that's only a gut-feeling.
Is it true? If it is, how do I prove such a thing?
2026-03-29 01:05:08.1774746308
Homomorphism between S4 and A4
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There is indeed no such homomorphism. An easy way to check this is to write down the size-$2$ subgroups of $S_4$, take the quotient of $S_4$ by them, and then observe that you never get $A_4$.
Alternatively, it suffices to prove that no size-$2$ subgroup of $S_4$ is normal. This is obvious because any normal subgroup containing a transposition must contain all transpositions!