Open AccessA note on groups generated by involutions and sharply 2-transitive groups
Author(s) -
George Glauberman,
Avinoam Mann,
Yoav Segev
Publication year - 2014
Publication title -
proceedings of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.968
H-Index - 84
eISSN - 1088-6826
pISSN - 0002-9939
DOI - 10.1090/s0002-9939-2014-12405-1
Subject(s) - annotation , algorithm , computer science , type (biology) , semantics (computer science) , artificial intelligence , mathematics , biology , programming language , ecology
Let G G be a group generated by a set C C of involutions which is closed under conjugation. Let π \pi be a set of odd primes. Assume that either (1) G G is solvable, or (2) G G is a linear group. We show that if the product of any two involutions in C C is a π \pi -element, then G G is solvable in both cases and G = O π ( G ) ⟨ t ⟩ G=O_{\pi }(G)\langle t\rangle , where t ∈ C t\in C . If (2) holds and the product of any two involutions in C C is a unipotent element, then G G is solvable. Finally we deduce that if G \mathcal {G} is a sharply 2 2 -transitive (infinite) group of odd (permutational) characteristic, such that every 3 3 involutions in G \mathcal {G} generate a solvable or a linear group; or if G \mathcal {G} is linear of (permutational) characteristic 0 , 0, then G \mathcal {G} contains a regular normal abelian subgroup.