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Let me quote some well-known results and perhaps related problems which may be illuminating! Let $G$ be a non-abelian finite $p$-group having cyclic center. Then, there is no finite $p$-group $H$ suc …

The intersection of all normalizers of subgroups of a group $G$ is called the norm of $G$. By a result of Schenkman [E. Schenkman, On the norm of a group, Illinois J. Math., 7 (1960) 150-152] the nor …

I think that a ``proper" question which must be proposed instead of Question 1, is the following: Is there a classification of finite $p$-groups $G$ such that both $G'$ and $Z(G)$ are cyclic? The la …

The answer is positive: since one must give a classification of at least one types of finite $p$-groups which I suggest to consider the classification of finite $p$-groups having a maximal subgroup w …

Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?

Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\beta$ is a non-zero element of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ such that $1\i …

Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ suc …

If $G$ is a non-trivial finite group which can be generated by two non-trivial elements of prime power orders, then the answer to the question is affirmative. Let $\mathcal{G}$ be the set of all subgr …

This is not a complete answer (only gives the answer for p=2,3,5) but it is also too long to add as a comment! Known results concerning similar questions as yours suggest that the nilpotency class of …