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On Primitive Extensions of Rank 3 of Symmetric Groups

Published online by Cambridge University Press:  22 January 2016

Tosiro Tsuzuku
Tosiro Tsuzuku*
Affiliation:
Nagoya University and University of Illinois
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1. Let Ω be a finite set of arbitrary elements and let (G, Ω) be a permutation group on Ω. (This is also simply denoted by G). Two permutation groups (G, Ω) and (G, Γ) are called isomorphic if there exist an isomorphism σ of G onto H and a one to one mapping τ of Ω onto Γ such that (g(i))τ=gσ(iτ) for g ∊ G and i∊Ω. For a subset Δ of Ω, those elements of G which leave each point of Δ individually fixed form a subgroup GΔ of G which is called a stabilizer of Δ. A subset Γ of Ω is called an orbit of GΔ if Γ is a minimal set on which each element of G induces a permutation. A permutation group (G, Ω) is called a group of rank n if G is transitive on Ω and the number of orbits of a stabilizer Ga of a ∊ Ω, is n. A group of rank 2 is nothing but a doubly transitive group and there exist a few results on structure of groups of rank 3 (cf. H. Wielandt [6], D. G. Higman M).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 171 - 177
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Burnside, W., Theory of groups of finite order, 2nd ed. Cambridge University Press, London, 1911.Google Scholar
[2] Frame, J. S., The double cosets of a finite group, Bull. Amer. Math. Soc, 47 (1941), 458–467.Google Scholar
[3] Frobenius, G., (i) Über die Charakter der symmetrishen Gruppe, Sitzber. Preuss. Akad., Berlin (1900), and, (ii) Über die Charaktere der alternierenden Gruppe, ibid. (1901), 303–315.Google Scholar
[4] Higman, D. G., Finite permutation group of rank 3, Math. Zeitshr. 86 (1964), 145–156.CrossRefGoogle Scholar
[5] Mitchell, H. H., Determination of the ordinary and modular ternary linear groups, Trans. Amer. Math. Soc. 12 (1911), 207–242.CrossRefGoogle Scholar
[6] Wielandt, H., Finite permutation groups, Academic Press, New York and London, 1964.Google Scholar