Let $G$ be a finite group and $\mathsf{cd}(G)$ denote the set of complex irreducible character degrees of $G$ . We prove that if $G$ is a finite group and $H$ is an almost simple group whose socle is a sporadic simple group $H_{0}$ and such that $\mathsf{cd}(G)=\mathsf{cd}(H)$ , then $G^{\prime }\cong H_{0}$ and there exists an abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$ . In view of Huppert’s conjecture, we also provide some examples to show that $G$ is not necessarily a direct product of $A$ and $H$ , so that we cannot extend the conjecture to almost simple groups.

We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph of some semigroup. Moreover, we obtain complete classifications of the graphs with an isolated vertex or edge that are the commuting graph of a group and the cycles that are the commuting graph of a centrefree semigroup.

We introduce the concept of infinite cochain sequences and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) and also how they can be applied to prove that almost all groups in such a family have equivalent Quillen categories. We also include some examples of infinite families of $p$ -groups from different coclass families that have equivalent Quillen categories.

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$ . We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$ ). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

The last term of the lower central series of a finite group $G$ is called the nilpotent residual. It is usually denoted by $\unicode[STIX]{x1D6FE}_{\infty }(G)$ . The lower Fitting series of $G$ is defined by $D_{0}(G)=G$ and $D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$ for $i=0,1,2,\ldots \,$ . These subgroups are generated by so-called coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ and $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ in elements of $G$ . More precisely, the set of coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ generates $\unicode[STIX]{x1D6FE}_{\infty }(G)$ whenever $k\geq 2$ while the set $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ generates $D_{k}(G)$ for $k\geq 0$ . The main result of this article is the following theorem: let $m$ be a positive integer and $G$ a finite group. Let $X\subset G$ be either the set of all $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ -commutators for some fixed $k\geq 2$ or the set of all $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ -commutators for some fixed $k\geq 1$ . Suppose that the size of $a^{X}$ is at most $m$ for any $a\in G$ . Then the order of $\langle X\rangle$ is $(k,m)$ -bounded.

Let $F$ be a field of characteristic $p\geq 0$ and $G$ any group. In this article, the Engel property of the group of units of the group algebra $FG$ is investigated. We show that if $G$ is locally finite, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G$ is locally nilpotent and $G^{\prime }$ is a $p$ -group. Suppose that the set of nilpotent elements of $FG$ is finite. It is also shown that if $G$ is torsion, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G^{\prime }$ is a finite $p$ -group and $FG$ is Lie Engel, if and only if ${\mathcal{U}}(FG)$ is locally nilpotent. If $G$ is nontorsion but $FG$ is semiprime, we show that the Engel property of ${\mathcal{U}}(FG)$ implies that the set of torsion elements of $G$ forms an abelian normal subgroup of $G$ .

Necessary and sufficient conditions for the existence of an orthogonal $\ast$ -basis of symmetry classes of tensors associated to nonabelian groups of order $pq$ are provided by using vanishing sums of roots of unity.

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$ . We determine the structure of $N$ when the diameter of the graph associated to the $G$ -conjugacy classes contained in $N$ is as large as possible, that is, equal to three.

In this paper we determine the structure of the total component of the Schur multiplier over an algebraically closed field of some relevant families of groups, such as dihedral groups, dicyclic groups, the infinite cyclic group and the direct product of two finite cyclic groups.

Let $p$ be an odd prime. In this note, we show that a finite group $G$ is solvable if all degrees of irreducible complex characters of $G$ not divisible by $p$ are either 1 or a prime.

We show that the proportion of permutations $g$ in $S_{\!n}$ or $A_{n}$ such that $g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality at most $\lceil n^{{\it\varepsilon}}\rceil$ is at least a constant multiple of ${\it\varepsilon}$ . Using this result, we obtain the same conclusion for elements in a classical group of natural dimension $n$ in odd characteristic that have even order and power up to an involution with $(-1)$ -eigenspace of dimension at most $\lceil n^{{\it\varepsilon}}\rceil$ for a linear or unitary group, or $2\lceil \lfloor n/2\rfloor ^{{\it\varepsilon}}\rceil$ for a symplectic or orthogonal group.

Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group $\text{A}_{39}$. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group $\text{A}_{40}$.

We classify all (saturated) fusion systems on bicyclic 2-groups. Here, a bicyclic group is a product of two cyclic subgroups. This extends previous work on fusion systems on metacyclic 2-groups. As an application we prove Olsson's conjecture for all blocks with bicyclic defect groups.

We define several graphs related to the p-blocks of a solvable group. We bound the diameter of these graphs when the defect group associated with the block is either abelian or normal and when the group has odd order. We give examples to show that these bounds are met.

Let $G$ be a finite group. We denote by ${\it\nu}(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup and by $\text{Nil}_{G}(x)$ the set of elements $y\in G$ such that $\langle x,y\rangle$ is a nilpotent subgroup. A group $G$ is called an ${\mathcal{N}}$-group if $\text{Nil}_{G}(x)$ is a subgroup of $G$ for all $x\in G$. We prove that if $G$ is an ${\mathcal{N}}$-group with ${\it\nu}(G)>\frac{1}{12}$, then $G$ is soluble. Also, we classify semisimple ${\mathcal{N}}$-groups with ${\it\nu}(G)=\frac{1}{12}$.

We prove the assertion in the title by conducting an exhaustive computational search for subgroups isomorphic to $\text{PSL}_{2}(13)$ and containing elements in class $13B$ .

Supplementary materials are available with this article.

Let $R$ be a commutative ring, let $F$ be a locally compact non-archimedean field of finite residual field $k$ of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. We show that the pro-$p$-Iwahori Hecke $R$-algebra of $G=\mathbf{G}(F)$ admits a presentation similar to the Iwahori–Matsumoto presentation of the Iwahori Hecke algebra of a Chevalley group, and alcove walk bases satisfying Bernstein relations. This was previously known only for a $F$-split group $\mathbf{G}$.

We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group $W$. The (two-colored) Temperley–Lieb category is embedded inside this category as the degree $0$ morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones–Wenzl projectors. When $W$ is infinite, the parameter $q$ of the Temperley–Lieb algebra may be generic, yielding a quantum version of the geometric Satake equivalence for $\mathfrak{sl}_{2}$. When $W$ is finite, $q$ must be specialized to an appropriate root of unity, and the negligible Jones–Wenzl projector yields the Soergel bimodule for the longest element of $W$.

A subset $X$ of a finite group $G$ is a set of pairwise noncommuting elements if $xy\neq yx$ for all $x\neq y\in X$. If $|X|\geq |Y|$ for any other subset $Y$ of pairwise noncommuting elements, then $X$ is called a maximal subset of pairwise noncommuting elements and the size of such a set is denoted by ${\it\omega}(G)$. In a recent article by Azad et al. [‘Maximal subsets of pairwise noncommuting elements of some finite $p$-groups’, Bull. Iran. Math. Soc.39(1) (2013), 187–192], the value of ${\it\omega}(G)$ is computed for certain $p$-groups $G$. In the present paper, our aim is to generalise these results and find ${\it\omega}(G)$ for some more $p$-groups of interest.