We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T ) of a maximal subgroup of T : for T not an alternating group we prove that, with finitely many exceptions, the maximum element order is at most m(T ). Moreover, apart from an explicit list of groups, the bound can be reduced to m(T )/4. These results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n/4. 2000 Mathematics Subject Classification. 20B15, 20H30.
We show that if G is a group of automorphisms of a thick finite generalised quadrangle Q acting primitively on both the points and lines of Q, then G is almost simple. Moreover, if G is also flag-transitive then G is of Lie type.
Two experiments were performed to examine (a) the encoding function of note-taking and (b) processing differences between successful and less successful students in lecture situations. In the first experiment, subjects either took notes or listened during a lecture. Different memory patterns were found for these two groups, with note-takers recalling many more high-importance propositions than low-importance propositions and non-note-takers recalling an equal number of high-and low-importance propositions. These results suggested that note-taking enhances organizational processing of lecture information. In the second experiment, the notes and recall of successful and less successful students were compared. Successful students recalled more of the most important propositions, but these two groups of students did not differ in their recall of less important propositions. For both groups of students, recall content was closely related to the content of the notes, with successful students recording more high-importance propositions in their notes. Also, successful and less successful students were quite similar in their note-taking styles and the degree to which they benefited from reviewing their notes. Thus, the memory difference between these two groups of students was interpreted to be the result of factors occurring during note-taking, with successful students engaging in greater integrative processing.
We prove that if Cay(G; S) is a connected Cayley graph with n vertices, and the prime factorization of n is very small, then Cay(G; S) has a hamiltonian cycle. More precisely, if p, q, and r are distinct primes, then n can be of the form kp with 24 = k < 32, or of the form kpq with k ≤ 5, or of the form pqr, or of the form kp 2 with k ≤ 4, or of the form kp 3 with k ≤ 2.
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