Let G be a primitive permutation group of finite degree n containing a subgroup H which fixes k points and has r orbits on Δ, the set of points it moves. An old and important theorem of Jordan says that if r = 1 and k ≥ 1 then G is 2-transitive; moreover if H acts primitively on Δ then G is (k + 1)-transitive. Three extensions of this result are proved here: (i) if r = 2 and k ≥ 2 then G is 2-transitive, (ii) if r = 2, n > 9 and H acts primitively on both of its two nontrivial orbits then G is k-primitive, (iii) if r = 3, n > 13 and H acts primitively on each of its three nontrivial orbits, all of which have size at least 3, then G is (k − 1)-primitive.

A Latin square is considered to be a set of n2 cells with three block systems. An automorphisni is a permutation of the cells which preserves each block system. The automorphism group of a Latin Square necessarily has at least 4 orbits on unordered pairs of cells if n < 2. It is shown that there are exactly 4 orbits if and only if the square is the composition table of an elementary abelian 2-group or the cclic group of order 3.

The purpose of this note is to determine the automorphism group of the doubly regular tournament of Szekeres type, and to use it to show that the corresponding skew Hadamard matrix H of order 2(q + 1), where q ≡5(mod 8) and q > 5, is not equivalent to the skew Hadamard matrix H(2q + 1) of quadratic residue type when 2q + 1 is a prime power.

Let G be a finite and u(G) the group of all invertible transformations (polynomial permutations) of the form x→a1 x1→ xk a2⃛ar xkr ar+1 (aiε G, x runs through G). Continuing investigations of H. Lausch of groups satisfying u(G) = {X→axk b} we show here that this condition implies that G is the direct product of its {2, 3}-Hall subgroup and its {2, 3}′-Hall subgroup H where H is nilpoint of class ≤2. Essentially all non-nilpoint groups G of order 2m 3n are described having the property u(G)= {x→axk b}

The proof of Theorem 5 in a paper with the same title is incorrect. In this note weaker versions of that theorem are proved.

Groups with the property of the title were considered by Chillag (1977); this paper completes his results by showing that, with known exceptions, they are triply transitive.

Let G be a doubly transitive permutation group on a finite set Ω, and let Kα be a normal subgroup of the stabilizer Gα of a point α in Ω. If the action of Gα on the set of orbits of Kα in Ω − {α} is 2-primitive with kernel Kα it is shown that either G is a normal extension of PSL(3, q) or Kα ∩ Gγ is a strongly closed subgroup of Gαγ in Gα, where γ ∈ Ω − {α}. If in addition the action of Gα on the set of orbits of Kα is assumed to be 3-transitive, extra information is obtained using permutation theoretic and centralizer ring methods. In the case where Kα has three orbits in Ω − {α} strong restrictions are obtained on either the structure of G or the degrees of certain irreducible characters of G. Subject classification (Amer. Math. Soc. (MOS) 1970: 20 B 20, 20 B 25.