A note on Sylow intersections

Ish-Shalom, Ariel

doi:10.1090/S0002-9939-1977-0460460-4

A note on Sylow intersections
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by Ariel Ish-Shalom PDF

Proc. Amer. Math. Soc. 66 (1977), 227-230 Request permission

Abstract:

Let p be a prime number, G a finite group whose order is divisible by p, and S a Sylow p-subgroup of G. We say that G satisfies $( \ast )$ if there exists $g \in G$ such that $S \cap {S^g} = {O_p}(G)$. There are examples of primes p and finite groups G that do not satisfy $( \ast )$. In this note we discuss several related properties satisfied in general, as well as giving sufficient conditions for G to satisfy $( \ast )$.

References

Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
Marcel Herzog, Intersections of nilpotent Hall subgroups, Pacific J. Math. 36 (1971), 331–333. MR 280595
Marcel Herzog, On Sylow intersections, Proc. Amer. Math. Soc. 37 (1973), 352–354. MR 310055, DOI 10.1090/S0002-9939-1973-0310055-4
Noboru Itô, Über den kleinsten $p$-Durchschnitt auflösbarer Gruppen, Arch. Math. (Basel) 9 (1958), 27–32 (German). MR 131455, DOI 10.1007/BF02287057
Thomas J. Laffey, A remark on minimal Sylow intersections, Bull. London Math. Soc. 4 (1972), 377. MR 318306, DOI 10.1112/blms/4.3.377
Thomas J. Laffey, On minimal Sylow intersections, J. London Math. Soc. (2) 12 (1975/76), no. 3, 383–384. MR 393226, DOI 10.1112/jlms/s2-12.3.383