Search Results

We know that the alternating group of degree $n>7$ has an irreducible character of degree $n-1$. The latter number is the smallest nontrivial one for each the alternating group has an irreducible char …

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 \;|\; …

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 …

Sylow $2$-subgroups of the symmetric group $S_n$ of degree $n$ are rational. There was a longstanding conjecture on rational groups saying that Sylow $2$-subgroups of a rational group are also ration …

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 …

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 …

Is there a group $G$ in which every abelian subgroup is finite and there is no upper bound on the sizes of its abelian subgroups? Let me say that a counterpart of the question above has posed by Paul …

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 $ …

The following conjecture/problem posed in The Kourovka Notebook in 1973 by Ya. G. Berkovich: Problem 4.13. Prove that every finite non-abelian $p$-group admits an automorphism of order $p$ which is …

Let $n>3$ be an odd integer and let $K_n$ denote the complete graph on $n$ vertices. For which integers $n$ the line graph $L(K_n)$ is a Cayley graph? For even $n$, it follows from a result of Watkins …

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

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 …

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 …

(1) Is there a finite nilpotent ring $R$ such that the ratio $$c(R)=\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$ is not integer? Edit 1: The nilpotent condition is put later. Edit/Answer: A …

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 …