Hiranya Kishore Dey

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming apartially ordered set. We prove a number of results about this poset, among them the following.1. M. Amiri recently proved that the poset has a unique maximal element, corresponding to the cyclic group.We show that the product of orders in a cyclic group of order $n$ is at least $q^{\phi(n)}$ times as large as the product in any non-cyclic group, where $q$ is the smallest prime divisor of $n$ and $\phi$ is Euler's function,with a similar result for the sum.2. The poset of order sequences of abelian groups of order $p^n$ is naturally isomorphic to the (well-studied) poset of partitions of $n$ with its natural partial order.3. If there exists a non-nilpotent group of order $n$, then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order $n$.4. There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups $G$ and $H$ is the order sequence of agroup if and only if $|G|$ and $|H|$ are coprime. The paper concludes with a number of open problems.

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a partially ordered set. We prove a number of results about this poset, among them the following. M.~Amiri recently proved that the poset has a unique maximal element, corresponding to the cyclic group. We show that the product of orders in a cyclic group of order $n$ is at least $q^{φ(n)}$ times as large as the product in any non-cyclic group,where $q$ is the smallest prime divisor of $n$ and $φ$ is Euler's function, with a similar result for the sum. The poset of order sequences of abelian groups of order $p^n$ is naturally isomorphic to the (well-studied) poset of partitions of $n$ with its natural partial order. If there exists a non-nilpotent group of order $n$, then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order $n$. There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups $G$ and $H$ is the order sequence of a group if and only if $|G|$ and $|H|$ are coprime. The paper concludes with a number of open problems.

The enhanced power graph of a group $G$ is the graph $\mathcal{G}_E(G)$ with vertex set $G$ and edge set $ \{(u,v): u, v \in \langle w \rangle,~\mbox{for some}~ w \in G\}$. In this paper, we compute the spectrum of the distance matrix of the enhanced power graph of non-abelian groups of order $pq$, dihedral groups, dicyclic groups, elementary abelian groups $\mathrm{El}(p^n)$ and the non-cyclic abelian groups $\mathrm{El}(p^n)\times\mathrm{El}(q^m)$ and $\mathrm{El}(p^n)\times \mathbb{Z}_m$, where $p$ and $q$ are distinct primes. For the non-cyclic abelian group $\mathrm{El}(p^n)\times \mathrm{El}(q^m)$, we also compute the spectrum of the adjacency matrix of its enhanced power graph and the spectrum of the adjacency and the distance matrix of its power graph.

Carlitz and Scoville in 1973 considered a four variable polynomial that enumerates permutations in $\mathfrak{S}_n$ with respect to the parity of its descents and ascents. In recent work, Pan and Zeng proved a $q$-analogue of Carlitz-Scoville's generating function by enumerating permutations with the above four statistice along with the inversion number. Further, they also proved a type B analogue by enumerating signed permutations with respect to the parity of descents and ascents. In this work we prove a $q$-analogue of the type B result of Pan and Zeng by enumerating permutations in $\mathfrak{B}_n$ with the above four statistics and the type B inversion number. We also obtain a $q$-analogue of the generating function for the type B bivariate alternating descent polynomials. We consider a similar five-variable polynomial in the type D Coxeter groups as well and give their egf. Alternating descents for the type D groups were previously also defined by Remmel, but our definition is slightly different. As a by-product of our proofs, we get bivariate $q$-analogues of Hyatt's recurrences for the type B and type D Eulerian polynomials. Further corollaries of our results are some symmetry relations for these polynomials and $q$-analogues of generating functions for snakes of types B and D.

We prove a new criterion for the solvability of the finite groups, depending on the function $\psi_k(G)$ which is defined as the sum of $k$-th powers of the element orders of $G$. We show that our result can be used to show the solvability of some groups for which the solvability does not follow from earlier similar kind of results and we emphasize the following: looking at $\psi_k(G)$ for $k>1$ can be useful to get further pieces of information about the group $G$.

The difference graph $D(G)$ of a finite group $G$ is the difference of enhanced power graph of $G$ and power graph of $G$, with all isolated vertices are removed. In this paper we study the connectedness and perfectness of $D(G)$ with respect to various properties of the underlying group $G$. We also find several connection between the difference graph of $G$ and the Gruenberg-Kegel graph of $G$.