On (1, 4)-Groups III

Citation

Buekenhout F & Rowlinson P (1976) On (1, 4)-Groups III. Journal of the London Mathematical Society, s2-14 (3), pp. 487-495. https://doi.org/10.1112/jlms/s2-14.3.487

Abstract
First paragraph: A (1, 4)-group is a transitive permutation group of even order in which the maximal number of points fixed by an involution is 4. Two previous papers, [10] and [3], were devoted to an investigation of the structure of Sylow 2-subgroups of a (1, 4)- group with no subgroup of index 2. Here we show that, with the natural exception of one "degenerate" class of (l,4)-groups (Theorem 3.4), a (l,4)-group of even permutations has sectional 2-rank at most 4 (equivalently, every 2-subgroup is 4-generated). Thus the recent work of Gorenstein and Harada [6] closes the gap between the conclusions of our own investigations and the hypotheses of an appropriate classification theorem. In particular, we are able to determine all the simple (l,4)-groups (§4). We are grateful to M. Aschbacher and J. Tits for providing information on the centralizers of involutions in the automorphism groups of known simple groups which has enabled us to identify the primitive (1, 4)-groups (§5).

Journal
Journal of the London Mathematical Society: Volume s2-14, Issue 3

People

Professor Peter Rowlinson

Emeritus Professor, Mathematics