TY - JOUR

T1 - A note on groups generated by involutions and sharply 2-transitive groups

AU - Glauberman, George

AU - Mann, Avinoam

AU - Segev, Yoav

N1 - Publisher Copyright:
© 2014 American Mathematical Society.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Let G be a group generated by a set C of involutions which is closed under conjugation. Let π be a set of odd primes. Assume that either (1) G is solvable, or (2) G is a linear group. We show that if the product of any two involutions in C is a π-element, then G is solvable in both cases and G = Oπ(G)〈t〉, where t ∈ C. If (2) holds and the product of any two involutions in C is a unipotent element, then G is solvable. Finally we deduce that if G is a sharply 2-transitive (infinite) group of odd (permutational) characteristic, such that every 3 involutions in G generate a solvable or a linear group; or if G is linear of (permutational) characteristic 0, then G contains a regular normal abelian subgroup.

AB - Let G be a group generated by a set C of involutions which is closed under conjugation. Let π be a set of odd primes. Assume that either (1) G is solvable, or (2) G is a linear group. We show that if the product of any two involutions in C is a π-element, then G is solvable in both cases and G = Oπ(G)〈t〉, where t ∈ C. If (2) holds and the product of any two involutions in C is a unipotent element, then G is solvable. Finally we deduce that if G is a sharply 2-transitive (infinite) group of odd (permutational) characteristic, such that every 3 involutions in G generate a solvable or a linear group; or if G is linear of (permutational) characteristic 0, then G contains a regular normal abelian subgroup.

UR - http://www.scopus.com/inward/record.url?scp=84923246453&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2014-12405-1

DO - 10.1090/S0002-9939-2014-12405-1

M3 - Article

AN - SCOPUS:84923246453

SN - 0002-9939

VL - 143

SP - 1925

EP - 1932

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 5

ER -