On subgroup structure of the p-group $G\simeq (Z/pZ)^{m}\rtimes(Z/pZ)$

Question

In order to study the subgroup structure of the p-group $G\simeq (Z/pZ)^{m}\rtimes(Z/pZ)$, I need to solve the following exercice from the Book (Dummit & Foote p101):

Exercice:

Let $H$ be a normal subgroup of $G$ of prime index $p$ then for all $K\leq G$ either

• $K\leq H$ or
• $G=HK$ and $|K:K\cap H|=p$.

Please refer to any other idea to define the subgroups of $G$. Any help would be appreciated so much. Thank you all.

Answer 1

Suppose that $H$ is a normal subgroup of $G$ of prime index. Moreover, assume that $ K$ is not a subgroup of $H$. Then, recall the following lemma:

If $H \le K \le G$ is a tower of groups. Then, $|G:H| = |G:K||K:H|$.

See Subsubgroups are subgroups of subgroups / Multiplicative Property of the Index for a discussion of this result.

Now, since $H$ is a normal subgroup, $HK$ is a subgroup of $G$. Hence, by the statement above we have that $|G:H| = |G:HK||HK:H|$ which implies that $p = |G:HK||HK:H|$. Since $p$ is prime $p \mid |G:HK|$ or $p | |HK:H|$, but not both. If $|HK:K| = 1$, then $K = HK$ which would mean $K \leq H$, contrary to our assumption. Hence, $|HK:K| = p$ which means that $|G:HK| = 1$, or $G = HK$. Now, apply the second isomorphism theorem to conclude that $|K:K\cap H| = p$.