1,510 research outputs found
Characterizations of Mersenne and 2-rooted primes
We give several characterizations of Mersenne primes (Theorem 1.1) and of
primes for which 2 is a primitive root (Theorem 1.2). These characterizations
involve group algebras, circulant matrices, binomial coefficients, and
bipartite graphs.Comment: 19 pages, final version, to appear in Finite Fields and their
Application
Base Size Sets and Determining Sets
Bridging the work of Cameron, Harary, and others, we examine the base size
set B(G) and determining set D(G) of several families of groups. The base size
set is the set of base sizes of all faithful actions of the group G on finite
sets. The determining set is the subset of B(G) obtained by restricting the
actions of G to automorphism groups of finite graphs. We show that for finite
abelian groups, B(G)=D(G)={1,2,...,k} where k is the number of elementary
divisors of G. We then characterize B(G) and D(G) for dihedral groups of the
form D_{p^k} and D_{2p^k}. Finally, we prove B(G) is not equal to D(G) for
dihedral groups of the form D_{pq} where p and q are distinct odd primes.Comment: 10 pages, 1 figur
Groups whose prime graphs have no triangles
Let G be a finite group and let cd(G) be the set of all complex irreducible
character degrees of G Let \rho(G) be the set of all primes which divide some
character degree of G. The prime graph \Delta(G) attached to G is a graph whose
vertex set is \rho(G) and there is an edge between two distinct primes u and v
if and only if the product uv divides some character degree of G. In this
paper, we show that if G is a finite group whose prime graph \Delta(G) has no
triangles, then \Delta(G) has at most 5 vertices. We also obtain a
classification of all finite graphs with 5 vertices and having no triangles
which can occur as prime graphs of some finite groups. Finally, we show that
the prime graph of a finite group can never be a cycle nor a tree with at least
5 vertices.Comment: 13 page
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