138 research outputs found
On the order sequence of a group
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 is the list of orders of elements of
the group, arranged in non-decreasing order. Order sequences of groups of order
are ordered by elementwise domination, forming a partially ordered set. We
prove a number of results about this poset, among them the following.
Abelian groups are uniquely determined by their order sequences, and the
poset of order sequences of abelian groups of order is naturally
isomorphic to the (well-studied) poset of partitions of with its natural
partial order.
If there exists a non-nilpotent group of order , then there exists such a
group whose order sequence is dominated by the order sequence of any nilpotent
group of order .
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 and is the order sequence of a group if and only if
and are coprime.
The paper concludes with a number of open problems.Comment: 22 pages, Comments are most welcom
Distance matrix of enhanced power graphs of finite groups
The enhanced power graph of a group is the graph with
vertex set 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 ,
dihedral groups, dicyclic groups, elementary abelian groups \El(p^n) and the
non-cyclic abelian groups \El(p^n)\times \El(q^m) and \El(p^n)\times
\mathbb{Z}_m, where and are distinct primes.
For the non-cyclic abelian group \El(p^n)\times \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
-enumeration of type B and D Eulerian polynomials based on parity of descents
Carlitz and Scoville in 1973 considered four-variable polynomials enumerating
the permutations according to the parity of both descents and ascents. In a
recent work, Pan and Zeng proved a -analogue of Carlitz-Scoville's
generating function by counting the inversion number. Moreover, they also
proved a type B analogue by enumerating the signed permutations with respect to
the parity of descent and ascent position. In this work we prove a -analogue
of the type B result of Pan and Zeng by counting the type B inversion number.
We also obtain a -analogue of the generating functions for the bivariate
alternating descent polynomials. Similar results are also obtained for type D
Coxeter groups. As a by-product of our proofs, we get -analogues of Hyatt's
recurrences for the type B and type D Eulerian polynomials.Comment: 23 pages. Comments are welcom
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