# The two-prime hypothesis: groups whose nonabelian composition factors are not isomorphic to $${{\mathrm{PSL}}}_2(q)$$ PSL 2 ( q )

The two-prime hypothesis: groups whose nonabelian composition factors are not isomorphic to... Let G be a finite group, and write $${\mathrm {cd}}(G)$$ cd ( G ) for the degree set of the complex irreducible characters of G. The group G is said to satisfy the two-prime hypothesis if, for any distinct degrees $$a, b \in {\mathrm {cd}}(G)$$ a , b ∈ cd ( G ) , the total number of (not necessarily different) primes of the greatest common divisor $$\gcd (a, b)$$ gcd ( a , b ) is at most 2. In this paper, we give an upper bound on the number of irreducible character degrees of nonsolvable groups satisfying the two-prime hypothesis and without composition factors isomorphic to $${{\mathrm{PSL}}}_2(q)$$ PSL 2 ( q ) for any prime power q. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals

# The two-prime hypothesis: groups whose nonabelian composition factors are not isomorphic to $${{\mathrm{PSL}}}_2(q)$$ PSL 2 ( q )

, Volume 184 (1) – Jul 18, 2016
17 pages

/lp/springer_journal/the-two-prime-hypothesis-groups-whose-nonabelian-composition-factors-X88bH95p2t
Publisher
Springer Vienna
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-016-0954-5
Publisher site
See Article on Publisher Site

### Abstract

Let G be a finite group, and write $${\mathrm {cd}}(G)$$ cd ( G ) for the degree set of the complex irreducible characters of G. The group G is said to satisfy the two-prime hypothesis if, for any distinct degrees $$a, b \in {\mathrm {cd}}(G)$$ a , b ∈ cd ( G ) , the total number of (not necessarily different) primes of the greatest common divisor $$\gcd (a, b)$$ gcd ( a , b ) is at most 2. In this paper, we give an upper bound on the number of irreducible character degrees of nonsolvable groups satisfying the two-prime hypothesis and without composition factors isomorphic to $${{\mathrm{PSL}}}_2(q)$$ PSL 2 ( q ) for any prime power q.

### Journal

Monatshefte f�r MathematikSpringer Journals

Published: Jul 18, 2016

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