L. BABAI, P. J. CAMERON, and P. P. P ALFY, On the orders of primitive groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161--168.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the class of finite groups all of whose nonabelian composition factors lie in S d (evidently, d includes all solvable groups) An earlier result of Luks [12] asserts that, for permutation groups in d , one can find set stabilizers in polynomial time (see also the work of Babai, Cameron, and Palfy [3] for important related results) In a future paper [16] we will generalize the methods of [13] that solve these graphisomorphism inspired matrix group problems from solvable groups to d (see also [17] The algorithms in [16] run deterministically in time polynomial in these three parameters: ....

L. BABAI, P. J. CAMERON, and P. P. P ALFY, On the orders of primitive groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161--168.

....(see, e.g. 19] for D, this could be the issue of conjugacy of two given subgroups. Manifestly, d includes all solvable groups. As indicated, the class arises naturally in significant instances of ISO. It has also become a subject of investigation in its own right asymptotic group theory (cf. [3], 25] The principal result of this paper is Theorem 1.1. Given permutation groups H G Sym( such that G 2 d , one can find the normalizer NormG (H) in polynomial time. By standard reductions, or by a careful review of the normalizer algorithm itself, one also derives a polynomial time ....

....for each of Problems A, B, and C, but extensions of these are required as well. We recall that the key property that resolved Problems A, B, and C was the fact that given any integer d 0 there is a function f(d) such that if a subgroup G of S n is primitive with G 2 d then jGj = O(n ) [3] (see also [16] This enabled a divide and conquer method that easily exploited orbits and, in the transitive case, used the primitive action on a block system to break the group into a small number of cosets of the intransitive stabilizer of the blocks. With care, and some additional tricks, ....

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L. BABAI, P. J. CAMERON, and P. P. P ALFY, On the orders of primitive groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161--168.

.... class of finite groups all of whose nonabelian composition factors lie in S d (evidently, # d includes all solvable groups) An earlier result of Luks [12] asserts that, for permutation groups in # d , one can find set stabilizers in polynomial time (see also the work of Babai, Cameron, and Palfy [3] for important related results) In a future paper [16] we will generalize the methods of [13] that solve these graphisomorphism inspired matrix group problems from solvable groups to # d (see also [17] The algorithms in [16] run deterministically in time polynomial in these three parameters: ....

L. BABAI, P. J. CAMERON, and P. P. P ALFY, On the orders of primitive groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161--168.

....not. For him this was a natural question in the context of properties of solvable groups he was studying. Orders of primitive subgroups G of Sn : If every noncyclic composition factor of G has order bounded by some constant b, then jGj = O(n c(b) for some constant c(b) Babai Cameron P alfy [BCP]) While this theorem certainly stands by itself, it was motivated by algorithmic applications. The classi cation of subgroups of GL(n; q) containing an irreducible element subject to suitable additional assumptions. Results of this type are discussed in the paper by Praeger in these ....

L. Babai, P. J. Cameron and P. Palfy, On the order of primitive groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161-168.

....of the form Sm acting on Gamma m k Delta , where k = k(m) log log m depends on m in an arbitrary way and for part 8 consider the group GL(m; q) acting on F m q , where q = q(m) log m depends on m in an arbitrary way. Part 9, follows from the following Theorem of Babai, Cameron and Palfy [BCP]. Theorem. Babai, Cameron and Palfy] For a n integer D, let G ae S n be a primitive 28 permutation group which does not involve A d as a section for d D. Then jGj is bounded by a polynomial in n (depending on D) This complete the proof of Theorem 3. Remark: The hypothesis of the Babai, ....

. L. Babai, P.J. Cameron and P.P. P'alfy, On the orders of primitive groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161--168.

....bounded orders. Furthermore, assuming a polynomial bound on the primes in jGj and jkj, it is then shown that, via a divide and conquer paradigm, in polynomial time one can solve VEC STAB and SUBSP STAB in d . These polynomial time algorithms exploit recent results in nite group theory such as [1]. My dissertation results will appear in articles [9] and [13] In solvable groups, there are also signi cant connections between SUBSP STAB and another entirely di erent generalization of SET STAB in permutation groups. The goal of my current project is to generalize such connections with the ....

L. Babai, P. J. Cameron, and P. P. P alfy, On the orders of primitive groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161-168.

....Then, in the joint work [GSSh] with Gluck and Seress, we show the following. Theorem 8 (Gluck Seress Shalev, 1998) Babai s conjecture holds. This provides a structural explanation for the celebrated Babai CameronP alfy theorem, stating that the order of the groups above is polynomial in n [BCP]. The original proof in [GSSh] yields f(d) O(d 2 ) A modified proof from [LiSh5] yields f(d) O(d) this implies the best bounds in the Babai CameronP alfy theorem, recently obtained by Pyber [P2] We also settle another base conjecture, posed by Cameron in [Ca] Conjecture 6 (Cameron, ....

L. Babai, P.J. Cameron and P.P. P'alfy, On the orders of primitive groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161-168.

....of whose noncyclic composition factors are isomorphic to a subgroup of Sym(d) in particular, Gamma d contains all solvable groups. A most significant effect of this restriction on a class of groups is that the primitive permutation groups (see [Wi] in the class have polynomially bounded order [BCP]. Primitive groups arise naturally as the base cases in certain divide and conquer procedures; see, e.g. Lu1] There are fairly elementary procedures for testing membership in Gamma d (see [Lu1, x4] For our purposes, it is essential only that d be fixed; the specific value of d would play ....

L. Babai, P. Cameron and P.J. P'alfy, On the order of primitive groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161-- 168.

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