Related papers: Limit points in the range of the commuting probabi…

For a finite group $G$, we associate the quantity $\beta(G)=\frac{|L(G)|}{|G|}$, where $L(G)$ is the subgroup lattice of $G$. Different properties and problems related to this ratio are studied throughout the paper. We determine the second…

Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$. Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where…

Motivated by analogous results for the symmetric group and compact Lie groups, we study the distribution of the number of fixed vectors of a random element of a finite classical group. We determine the limiting moments of these…

Let $L/K$ be a Galois extension of number fields and let $G=\mathrm{Gal}(L/K)$. We show that under certain hypotheses on $G$, for a fixed prime number $p$, Leopoldt's conjecture at $p$ for certain proper intermediate fields of $L/K$ implies…

Let $G$ be a finite group and $\Aut(G)$ the automorphism group of $G$. The autocommuting probability of $G$, denoted by $\Pr(G, \Aut(G))$, is the probability that a randomly chosen automorphism of $G$ fixes a randomly chosen element of $G$.…

H\'ethelyi and K\"ulshammer showed that the number of conjugacy classes $k(G)$ of any solvable finite group $G$ whose order is divisible by the square of a prime $p$ is at least $(49p+1)/60$. Here an asymptotic generalization of this result…

We say that a finite subset of the unit sphere in $\mathbf{R}^d$ is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times $(\log…

Let $G$ be a finite transitive group on a set $\Omega$, let $\alpha\in \Omega$ and let $G_\alpha$ be the stabilizer of the point $\alpha$ in $G$. In this paper, we are interested in the proportion $$\frac{|\{\omega\in \Omega\mid \omega…

Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\rm Isolated}(G)$ the set of isolated subgroups of $G$. In this note, we describe finite groups $G$ such that $|{\rm Isolated}(G)|=|L(G)|-k$, where…

The 'degree of k-step nilpotence' of a finite group G is the proportion of the tuples (x_1,...,x_{k+1}) in G^{k+1} for which the simple commutator [x_1,...,x_{k+1}] is equal to the identity. In this paper we study versions of this for an…

Let $G$ be a transitive permutation group acting on a finite set $\Omega$ with $|\Omega|\geqslant 2$. An element of $G$ is said to be a derangement if it has no fixed points on $\Omega$, and by a theorem of Jordan from 1872, $G$ always…

Let $E$ be an elliptic curve of rank $\text{rk}(E) \geq 1$, and let $E(\mathbb{F}_p)$ be the elliptic group of order $\#E(\mathbb{F}_p)=n$. The number of primes $p\leq x$ such that $n$ is prime is expected to be $\pi(x,E)=\delta(E)x/\log^2…

We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if $k$ is a positive integer such that for any prime $p$ the number of character codegrees of a finite…

A finite group G is exceptional if it has a quotient Q whose minimal faithful permutation degree is greater than that of G. We say that Q is a distinguished quotient. The smallest examples of exceptional p-groups have order p^5. For an odd…

Let $G$ be a finite group, define $I(G)=\{x\in G : x^{2}=1\}$, $C(G)=$ set of the cyclic subgroups of $G$, $i(G)=|I(G)|$ and $c(G)=|C(G)|$. In this article, we will classify finite groups with $i(G)=c(G)-r$ for $r=0,1,$ and $2$. We also…

Let $q$ be a power of a prime $p$ and let $G$ be a completely reducible subgroup of $\mathrm{GL}(d,q)$. We prove that the number of composition factors of $G$ that have prime order $p$ is at most $(\varepsilon_q d-1)/(p-1)$, where…

The so--called subgroup commutativity degree $sd(G)$ of a finite group $G$ is the number of permuting subgroups $(H,K) \in \mathrm{L}(G) \times \mathrm{L}(G)$, where $\mathrm{L}(G)$ is the subgroup lattice of $G$, divided by…

We say that a finite group $G$ acting on a set $\Omega$ has Property $(*)_p$ for a prime $p$ if $P_\omega$ is a Sylow $p$-subgroup of $G_\omega$ for all $\omega\in\Omega$ and Sylow $p$-subgroups $P$ of $G$. Property $(*)_p$ arose in the…

If $G$ is a finite group and $k =q>2$ or $k=q+1$ for a prime power $q$ then, for infinitely many integers $v$, there is a $2$-$(v,k,1)$-design ${\bf D}$ for which ${\rm Aut} {\bf D}\cong G$.

We prove that any multiplicative subgroup G of the prime field f_p with |G| < p^{1/2} satisfies |3G| \gg |G|^2 / \log |G|. Also, we obtain a bound for the multiplicative energy of any nonzero shift of G, namely E^* (G+x) \ll |G|^2 log |G|,…