Related papers: On the Exponent Conjectures

Let $G$ be an almost simple group. We prove that if $x \in G$ has prime order $p \ge 5$, then there exists an involution $y$ such that $<x,y>$ is not solvable. Also, if $x$ is an involution then there exist three conjugates of $x$ that…

Let $G$ be a nontrivial permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is…

Let $p^k m^2$ be an odd perfect number with special prime $p$. In this article, we provide an alternative proof for the biconditional that $\sigma(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We then give an application…

For a prime $p,$ let $\mathbb{F}_q$ be a finite extension of $\mathbb{F}_p.$ The restriction of an irreducible mod $p$ representation of $\text{GL}_2(\mathbb{F}_q)$ to its subgroup $\text{GL}_2(\mathbb{F}_p)$ can be seen as a tensor product…

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…

The minimal faithful permutation degree $\mu(G)$ of a finite group $G$ is the least integer $n$ such that $G$ is isomorphic to a subgroup of the symmetric group $S_n$. If $G$ has a normal subgroup $N$ such that $\mu(G/N) > \mu(G)$, then $G$…

In this paper we present two new results on the number of certain conjugacy classes of a finite group. For a finite group $G$, let $n(G)$ be the maximum of $k_{p}(G)$ taken over all primes $p$ where $k_{p}(G)$ denotes the number of…

Let $p<q$ be odd primes, $\rho_1$ and $\rho_2$ be irreducible representations of $\text{SL}(2,\mathbb{Z}_p)$ and $\text{SL}(2,\mathbb{Z}_q)$ of dimensions $\frac{p+1}{2}$ and $\frac{q+1}{2}$, respectively. We show that if…

A subgroup $H$ of a group $G$ is said to be {pronormal} in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. Some problems in finite group theory, combinatorics, and permutation group theory were solved in…

Let $p$ be a prime and let $S_2(\Gamma(p))$ be the space of weight $2$ cusp forms for the principal congruence subgroup $\Gamma(p)$. Then $\mathrm{SL}_2(\mathbb{F}_p)$ acts on $S_2(\Gamma(p))$ in a natural way. Around 1928, Hecke proved…

Let $p$ be a prime number, $G$ be a $p$-solvable finite group and $P$ be a Sylow $p$-subgroup of $G$. We prove that $G$ is $p$-supersolvable if $N_G(P)$ is $p$-supersolvable and if there is a subgroup $H$ of $P$ with $P' \le H \le \Phi(P)$…

Let $G$ be a finite group and $\psi(G) = \sum_{g \in G} o(g)$, where $o(g)$ denotes the order of $g \in G$. First, we prove that if $G$ is a group of order $n$ and $\psi(G) >31\psi(C_n)/77$, where $C_n$ is the cyclic group of order $n$,…

Let $p$ be an odd prime. Define the Gaussian power sum \[ G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n\in\mathbb Z[i]. \] We determine $G_p(p)$ modulo high powers of $p$: if $p\equiv 1\pmod 4$ then $$G_p(p)\equiv p^2(1+i)\pmod{p^3},$$…

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $\lambda (G)$ as the minimum number of nonsoluble factors in a series of…

Let $G$ be a finite $p$-group, where $p$ is an odd prime number, $H$ be a subgroup of $G$ and $\theta\in \Irr(H)$ be an irreducible character of $H$. Assume also that $|G:H|=p^2$. Then the character $\theta^G$ of $ G$ induced by $\theta$ is…

We prove the $p$-part of the strong Stark conjecture for every totally odd character and every odd prime $p$. Let $L/K$ be a finite Galois CM-extension with Galois group $G$, which has an abelian Sylow $p$-subgroup for an odd prime $p$. We…

For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has…

Let $d$ be an odd square-free integer, $m\geq 3$ any integer and $L_{m, d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic…

Let G be a finite group, Irr(G) the set of all irreducible complex characters of G and X \in Irr(G). Let also cod(X) = |G : kerX|/X(1) and cod(G) = {cod(X) | X \in Irr(G)}. In this note, we show that the simple group PSL(2, q), for a prime…

We give a computationally effective criterion for determining whether a finite-index subgroup of SL(2, Z) is a congruence subgroup, extending earlier work of Hsu for subgroups of PSL(2, Z).