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# Group Homomorphism: Identifying Kernels by Their Index

ABSALG-7TNKXD

Let $G$ be a group and $H$ a subgroup of $G$.

Which of the following properties does not always imply that $H$ is the kernel of some group homomorphism of $G$?

A

$[G:H]=2$

B

$[G:H]=3$ and $G$ has even order.

C

$[G:H]=3$ and $G$ has odd order divisible by $3$.

D

$|G|=35$ and $[G:H]=5$