In finite group theory, a p-stable group for an odd prime p is a finite group satisfying a technical condition introduced by Gorenstein and Walter in order to extend Thompson's uniqueness results in the odd order theorem to groups with dihedral Sylow 2-subgroups. There are several equivalent definitions of a p-stable group. First definition. We give definition of a p-stable group in two parts; the definition used here comes from. 1. Let p be an odd prime and G be a finite group with a nontrivial p-core O p. G is p-stable if it satisfies the following condition: Let P be an arbitrary p-subgroup of G such that O p ′ is a normal subgroup of G. Suppose that x ∈ N G and x ¯ is the coset of C G containing x. If = 1 x ¯ ∈ O n. Now, define M p as the set of all p-subgroups of G maximal with respect to the property that O p ≠ 1. 2. Let G be a finite group and p an odd prime. G is called p-stable if every element of M p is p-stable by definition 1. Second definition. Let p be an odd prime and H a finite group.
H is p-stable if F ∗ = O p and, whenever P is a normal p-subgroup of H and g ∈ H with = 1 g C H ∈ O p. If p is an odd prime and G is a finite group such that SL2 is not involved in G G is p-stable. If furthermore G contains a normal p-subgroup P such that C G ⩽ P Z is a characteristic subgroup of G, where J 0 is the subgroup introduced by John Thompson in. P-stability is used as one of the conditions in Glauberman's ZJ theorem. Quadratic pair p-constrained group p-solvable group Glauberman, George, "A characteristic subgroup of a p-stable group", Canadian Journal of Mathematics, 20: 1101–1135, doi:10.4153/cjm-1968-107-2, ISSN 0008-414X, MR 0230807 Thompson, John G. "A replacement theorem for p-groups and a conjecture", Journal of Algebra, 13: 149–151, doi:10.1016/0021-869390068-4, ISSN 0021-8693, MR 0245683 Gorenstein, D.. "On the maximal subgroups of finite simple groups", Journal of Algebra, 1: 168–213, doi:10.1016/0021-869390032-8, ISSN 0021-8693, MR 0172917 Gorenstein, D.. "The characterization of finite groups with dihedral Sylow 2-subgroups.
I", Journal of Algebra, 2: 85–151, doi:10.1016/0021-869390027-X, ISSN 0021-8693, MR 0177032 Gorenstein, D.. "The characterization of finite groups with dihedral Sylow 2-subgroups. II", Journal of Algebra, 2: 218–270, doi:10.1016/0021-869390019-0, ISSN 0021-8693, MR 0177032 Gorenstein, D.. "The characterization of finite groups with dihedral Sylow 2-subgroups. III", Journal of Algebra, 2: 354–393, doi:10.1016/0021-869390015-3, ISSN 0021-8693, MR 0190220 Gorenstein, D. "The classification of finite simple groups. I. Simple groups and local analysis", American Mathematical Society. Bulletin. New Series, 1: 43–199, doi:10.1090/S0273-0979-1979-14551-8, ISSN 0002-9904, MR 0513750 Gorenstein, D. Finite groups, New York: Chelsea Publishing Co. ISBN 978-0-8284-0301-6, MR 0569209
The Men's Greco-Roman 57 kg at the 1980 Summer Olympics as part of the wrestling program were held at the Athletics Fieldhouse, Central Sports Club of the Army. The competition used a form of negative points tournament, with negative points given for any result short of a fall. Accumulation of 6 negative points eliminated the loser wrestler; when only three wrestlers remain, a special final round is used to determine the order of the medals. LegendTF — Won by Fall IN — Won by Opponent Injury DQ — Won by Passivity D1 — Won by Passivity, the winner is passive too D2 — Both wrestlers lost by Passivity FF — Won by Forfeit DNA — Did not appear TPP — Total penalty points MPP — Match penalty pointsPenalties0 — Won by Fall, Technical Superiority, Passivity and Forfeit 0.5 — Won by Points, 8-11 points difference 1 — Won by Points, 1-7 points difference 2 — Won by Passivity, the winner is passive too 3 — Lost by Points, 1-7 points difference 3.5 — Lost by Points, 8-11 points difference 4 — Lost by Fall, Technical Superiority, Passivity and Forfeit Results from the preliminary round are carried forward into the final.