TY - JOUR

T1 - A new solvability criterion for finite groups

AU - Dolfi, Silvio

AU - Guralnick, Robert M.

AU - Herzog, Marcel

AU - Praeger, Cheryl E.

N1 - Funding Information:
The first author was partially supported by the MIUR project ‘Teoria dei gruppi e applicazioni’. The second author was supported by NSF grant DMS 1001962. The fourth author was supported by Federation Fellowship FF0776186 of the Australian Research Council.

PY - 2012/4

Y1 - 2012/4

N2 - In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x∈C and y∈D for which 〈x, y〉 is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y∈G with |x|=a and |y|=b, the subgroup 〈x, y〉 is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.

AB - In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x∈C and y∈D for which 〈x, y〉 is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y∈G with |x|=a and |y|=b, the subgroup 〈x, y〉 is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.

UR - http://www.scopus.com/inward/record.url?scp=84858391006&partnerID=8YFLogxK

U2 - 10.1112/jlms/jdr041

DO - 10.1112/jlms/jdr041

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AN - SCOPUS:84858391006

SN - 0024-6107

VL - 85

SP - 269

EP - 281

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 2

ER -