Name
Abstract | ||
|---|---|---|
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 is an element of C and y is an element of 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 vertical bar G vertical bar such that, for all elements x, y is an element of G with vertical bar x vertical bar=a and vertical bar y vertical bar=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. |
Year | DOI | Venue |
|---|---|---|
2012 | 10.1112/jlms/jdr041 | JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES |
Keywords | Field | DocType |
solvable group,simple group,conjugacy class,group theory | Topology,Feit–Thompson theorem,Primitive permutation group,Combinatorics,Subgroup,Mathematical analysis,p-group,Fitting subgroup,Solvable group,Finite group,Mathematics,Simple group | Journal |
Volume | Issue | ISSN |
85 | 2 | 0024-6107 |
Citations | PageRank | References |
2 | 0.86 | 1 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Silvio Dolfi | 1 | 2 | 0.86 |
| Robert M. Guralnick | 2 | 11 | 3.65 |
| Marcel Herzog | 3 | 9 | 4.66 |
| Cheryl E. Praeger | 4 | 545 | 100.88 |