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There are several results in the literature concerning $p$-groups $G$ with amaximal elementary abelian normal subgroup of rank $k$ due to Thompson, Mannand others. Following an idea of Sambale we obtain bounds for the number ofgenerators etc. of a $2$-group $G$ in terms of $k$, which were previously knownonly for $p>2$. We also prove a theorem that is new even for odd primes.Namely, we show that if $G$ has a maximal elementary abelian normal subgroup ofrank $k$, then for any abelian subgroup $A$ the Frattini subgroup $\Phi(A)$ canbe generated by $2k$ elements ($3k$ when $p=2$). The proof of this rests uponthe following result of independent interest: If $V$ is an $n$-dimensionalvector space, then any commutative subalgebra of End$(V)$ contains a zeroalgebra of codimension at most $n$.

On $p$-groups with a maximal elementary abelian normal subgroup of rank $k$