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Is it true that a pro-2 group $G$ with $G/\Phi(G)\cong C_2\times C_2$ may have a maximal subgroup which is not open? Motivation: The Frattini subgroup of a profinite group by definition, is the inter …

Let the group $G$ have the presentation $\langle x_1, \dots, x_n \;|\; r_1, \dots, r_m \rangle$. Let $\Gamma$ be a labelled directed graph corresponding to Van Kampen diagram over the above presentat …

Are there a finite group $G$ and a field $\mathbb{F}$ such that $\gcd(3,|G|)=1$ and the group algebra $\mathbb{F}[G]$ contains a zero divisor whose support is of size $3$? Recall that the support of …

A group $G$ is called of generalized exponent $n$ if there exists elements $a_1,\dots,a_n \in G$ such that $x^{a_1}\cdots x^{a_n}=1$ for all $x\in G$, where $x^a=a^{-1}xa$. See the following question …

Is there an infinite finitely generated group whose Frattini factor is isomorphic to Klein 4-group?

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 …

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 …

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 …

It is called a power automorphism of the group $G$. The automorphism $g$ maps every element $x$ of $G$ to a power of $x$. See the following reference as a starting point. Christopher D. H. Cooper, Po …

Search morphic groups. Some papers of Y. Li from Brock University. EDIT: Thanks to Misha. Let me recall the definition of a morphic group. A group $G$ is called morphic if every endomorphism $\alpha …

A finite group $G$ is called rational if and only if $N_G(\langle x\rangle)/C_G(x)\cong Aut(\langle x\rangle)$ for all $x\in G$. The word ``rational" is because there is an equivalent definition in gr …

The solution of the restricted Burnside problem is based on the Hall - Higman reduction, the classification of finite simple groups and a celebrated result of Efim Zelmanov on finding an upper bound f …

Let $G_d$ be the group with the following presentation $$\langle x,y \mid x^{2^{d+1}}=1, x^4=y^2, [x,y,x]=x^{2^{d}}, [x,y,y]=1\rangle,$$ where $d>2$ is an integer. It is clear that $G_d$ is a finite $ …

Let $G$ be a finite supersolvable group with trivial Frattini subgroup. Is it true that all Sylow subgroups of $G$ are elementary abelian? EDIT: Many Thanks to Derek for his answer. Let me say some w …

Is there a characterization of finite groups having no dihedral subgroup of order $2p$ for all odd primes $p$ dividing the order of the group?