On Finite Groups of Even Order Whose 2-Sylow Group Is a Quaternion Group

3-Nets realizing a group in a projective plane

“…Every inflection point of a nonsingular cubic F is the center of an involutory homology preserving F . A classical Lame configuration consists of two triples of distinct lines in PG(2, K), say 1 , 2 , 3 and r 1 , r 2 , r 3 , such that no line from one triple passes through the common point of two lines from the other triple. For 1 ≤ j, k ≤ 3, let R jk denote the common point of the lines j and r k .…”

“…Doing so, three points u 1 ∈ Λ 1 , v 2 ∈ Λ 2 , w 3 ∈ Λ 3 are collinear if and only if uv = w in D n . Therefore, for 1 ≤ i ≤ n − 2, the triangle with vertices x 2 , (xy) 2 , (xy −i ) 3 and that with vertices (1) 3 , y 3 , (y −i ) 2 are in mutual perspective position from the point x 1 . For two distinct points u i and v j with u i ∈ Λ i and v j ∈ Λ j and 1 ≤ i, j ≤ 3, let u i v j denote the line through u i and v j .…”

“…Then for all x ∈ G, ψ 1 interchanges the points x 1 and x 3 . The point a 2 ∈ Λ 2 is the intersection of the lines y 1 (ya) 3 , with y ∈ G. These lines are mapped to the lines (ya) 1 y 3 , which all contain the point (a −1 ) 2 of Λ 2 . Therefore, the involutory homology ψ 1 leaves Λ 2 invariant.…”

“…Therefore, |G| = 24. Two possibilities arise according as either G ∼ = SL (2,3) or G is the dicyclic group of order 24. The latter case cannot actually occur by Proposition 30 as the dicyclic group of order 24 has a (normal) cyclic subgroup of order 12.…”

“…To finish the proof, it suffices to observe that SL(2, q h ), with q h as before, contains SL(2, 3), whereas no dual 3-net can realize SL(2, 3) as we have already pointed out. Proof Let G be a non-Abelian simple group, and consider a Sylow 2-subgroup S 2 of G. By Proposition 31, S 2 is dihedral since no Sylow 2-subgroup of a non-Abelian simple group is either cyclic or the direct product of cyclic groups, see [5, Theorem 2.168], or the quaternion group of order 8, see [3]. From the Gorenstein-Walter theorem [6], either G ∼ = PSL (2, q h ) with an odd prime q and q h ≥ 5, or G ∼ = Alt 7 .…”