Let $G$ be a group with 81 elements and $H$ a subgroup with 27 elements. Which of the following is not true?
- (a) $H$ is a normal subgroup of $G$
- (b) $Z(H)\neq 1$
- (c) $H'=1$
- (d) $G'\subseteq H$
I know that (b) is true. Since the center of every nontrivial p-group is nontrivial (p is prime). Also, (d) is true since $G/H$ has 3 elements and hence $G/H$ is abelian. So $G'\subseteq H$. But (a) and (c)?