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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 …

The Fischer graph is one of examples; see page 569 of Suzuki, Michio. Group theory. II. Translated from the Japanese. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemati …

Let $c_1,\dots,c_k$ be some non-zero complex numbers and $M$ be the abelian subgroup generated by $c_1,\dots,c_k$ (i.e. all $\mathbb{Z}$-linear combinations of $c_1\dots,c_k$). Suppose further that $\ …

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 …

For Quesion 5: What is known today about normal closures of elements in n-Engel groups? see the following papers: Traustason, Gunnar, Locally nilpotent 4-Engel groups are Fitting groups. J. Algebr …

See the followings: Malcev, A. I. On a general method for obtaining local theorems in group theory, Notices of the Pedagogical Institute of Ivanovo, Physical-Mathematical Sciences, 1, 3-9 (in R …

Problem 1.12 of [Unsolved Problems in Group Theory, The Kourovka Notebook, Novosibirsk, 2010]: (W. Magnus) The problem of the isomorphism to the trivial group for all groups with $n$ generators and …

Every word $w$ on free generators $x_1,\dots,x_n$ can be written as $$w=x_1^{\alpha_1} \cdots x_n^{\alpha_n} c(x_1,\dots,x_n),$$ where $c$ is a word in the commutator subgroup of $\langle x_1,\dots,x …

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 …

The number $k(G)$ is the domination number of the non-commuting graph of $G$. See Proposition 2.14 of [J. Algebra, 298 (2006) 468–492]. By Corollary 2.17 of [J. Algebra, 298 (2006) 468–492], if $k(H …

Recall that a discrete group $G$ is amenable if and only if it has Folner condition, i.e., for every positive number $\epsilon$ and every finite set $A$ of $G$ there exists a finite non-empty subset …

Is there a torsion-free group $G=\langle x,y \rangle$ such that the directed Cayley graph $\Gamma=Cay(G,\{x,y,x^{-1}y\})$ contains a finite cubic induced subgraph? The vertex set of $\Gamma$ is $G$ an …

Is there a finite non-abelian $2$-group $G$ without non-trivial elementary abelian direct factor and of order $2^9$ satisfying the following condition: $$Z(G) \cap Z(\Phi(G))= \langle \prod_{i=1}^{2^d …

Let $G_1$ and $G_2$ be the groups with the following presentations: $$G_1=\langle a,b \;|\; (ab)^2=a^{-1}ba^{-1}, (a^{-1}ba^{-1})^2=b^{-2}a, (ba^{-1})^2=a^{-2}b^2 \rangle,$$ $$G_2=\langle a,b \;|\; …