Base sizes of primitive permutation groups

This work was supported by: EPSRC Grant Numbers EP/R014604/1 and EP/M022641/1.Let G be a permutation group, acting on a set Ω of size n. A subset B of Ω is a base for G if the pointwise stabilizer G(B) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of Sym(n) is large base if there exist integers m and r ≥ 1 such that Alt (m)r ... G ≤ Sym (m) \wr Sym (r), where the action of Sym (m) is on k-element subsets of {1,...,m} and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group M24 in its natural action on 24 points, or b(G) ≤ ⌈log n⌉ + 1. Furthermore, we show that there are infinitely many primitive groups G that are not large base for which b(G) > log n + 1, so our bound is optimal.Publisher PDFPeer reviewe

Groups satisfying a strong complement property

Let G = NH be a finite group where N is normal in G and H is a complement of N in G. For a given generating sequence (h(1),.., h(d)) of H we say that (N, (h(1),..., h(d))) satisfies the strong complement property, if is a complement of N in G for all x(1),..., x(d) is an element of N. When d is the minimal number of elements needed to generate H, and (N, (h(1),..., h(d)>)) satisfies the strong complement property for every generating sequence (h(1),..., h(d)) with length d, then we say that (N, H) satisfies the strong complement property. In the case when vertical bar N vertical bar and vertical bar H vertical bar are coprime, we show that (N, H) can only satisfy the strong complement property if H is cyclic or if H acts trivially on N. We give on the other hand a number of examples that show this does not need to be the case when considering the strong complement property of (N, (h(1),..., h(d))) for a given fixed generating sequence. In the case when N and H are not of coprime order, we give examples where (N, H) satisfies the strong complement property and where H is not cyclic and does not act trivially on N

Boolean lattices in finite alternating and symmetric groups

Given a group $G$ and a subgroup $H$, we let $\mathcal{O}_G(H)$ denote the lattice of subgroups of $G$ containing $H$. This paper provides a classification of the subgroups $H$ of $G$ such that $\mathcal{O}_{G}(H)$ is Boolean of rank at least $3$, when $G$ is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilizers of chains of regular partitions, and the other type arises by taking stabilizers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices, related to the dual Ore's theorem and to a problem of Kenneth Brown.Comment: 25 pages, classification of Boolean lattices in symmetric and alternating group

The expected number of random elements to generate a profinite group

Let G be a prosolvable group with all the Sylow subgroups d-generated. We give the best upper bound for the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators of G is found. Moreover we find an upper bound to this value near to the best possible for all the profinite groups having all the quotient of odd order and all the Sylow subgroups d-generated. Finally, let G be a permutation group of degree n>15, we give the best upper bound for the expected number of element of G needed to generate G itself

Comparing the expected number of random elements from the symmetric and the alternating groups needed to generate a transitive subgroup

Given a transitive permutation group of degree n, we denote by e(T) (G) the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators of a transitive subgroup of G is found. We compare e(T) (Sym(n)) and e(T) (Alt(n))

The expected number of elements to generate a finite group with d-generated sylow subgroups

Given a finite group G; let e(G) be the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators is found. If all of the Sylow subgroups of G can be generated by d elements, then e(G) 64 d + \u3ba, where \u3ba is an absolute constant that is explicitly described in terms of the Riemann zeta function and is the best possible in this context. Approximately, \u3ba equals 2.752394. If G is a permutation group of degree n; then either G = Sym(3) and e(G) = 2:9 or e(G) 64 {n=2}+\u3ba with \u3ba 3c1:606695: These results improve weaker bounds recently obtained by Lucchini

Base sizes of primitive permutation groups

Let G be a permutation group, acting on a set Ω of size n. A subset B of Ω is a base for G if the pointwise stabilizer G(B) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of Sym(n) is large base if there exist integers m and r ≥ 1 such that Alt (m)r ... G ≤ Sym (m) \wr Sym (r), where the action of Sym (m) is on k-element subsets of {1,...,m} and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group M24 in its natural action on 24 points, or b(G) ≤ ⌈log n⌉ + 1. Furthermore, we show that there are infinitely many primitive groups G that are not large base for which b(G) > log n + 1, so our bound is optimal