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Let $G$ be a group with a normal subgroup $N$ and a subgroup $H$ with $|H|=p$ of prime cardinality and $H$ is not a subgroup of $N$ . The question is to prove that $G/N$ has a subgroup that is isomorphic to $H$ . My attempt : $|H|=pimplies $ $H$ is cyclic. $H=langle hrangle$ for some $hin H$ The correspondance theorem states that $f:{Kle G : Ksupset N}rightarrow{text{subgroups of } G/N}$ is a bijection but this doesn't seem to be useful here. A hint would be welcome
abstract-algebra group-theory
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asked Nov 28 '18 at 11:51
John Cataldo
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