The Sylow subgroups of the symmetric group

Maximal 2-Local Geometries for the Symmetric Groups

“…; O r denote these orbits where jO i j ¼ 2 n i for i 2 I. Note that T ¼ T 1 Â T 2 Â Á Á Á Â T r where, T i 2 Syl 2 ðSymðO i ÞÞ; i 2 I and, moreover, each T i is a certain iterated wreath product of cyclic groups of order 2 (see Findlay, 1904;Kaloujnine, 1948;Weir, 1955or Proposition 19.10 of Humphreys, 1996. Further each T i (provided n i 6 ¼ 0) leaves invariant a (unique) block system of O i in which the blocks have size 2.…”