3,561 research outputs found
A note on the probability of generating alternating or symmetric groups
We improve on recent estimates for the probability of generating the
alternating and symmetric groups and . In
particular we find the sharp lower bound, if the probability is given by a
quadratic in . This leads to improved bounds on the largest number
such that a direct product of copies
of can be generated by two elements
Constructive homomorphisms for classical groups
Let Omega be a quasisimple classical group in its natural representation over
a finite vector space V, and let Delta be its normaliser in the general linear
group. We construct the projection from Delta to Delta/Omega and provide fast,
polynomial-time algorithms for computing the image of an element. Given a
discrete logarithm oracle, we also represent Delta/Omega as a group with at
most 3 generators and 6 relations. We then compute canonical representatives
for the cosets of Omega. A key ingredient of our algorithms is a new,
asymptotically fast method for constructing isometries between spaces with
forms. Our results are useful for the matrix group recognition project, can be
used to solve element conjugacy problems, and can improve algorithms to
construct maximal subgroups
Coprime invariable generation and minimal-exponent groups
A finite group is \emph{coprimely-invariably generated} if there exists a
set of generators of with the property that the orders
are pairwise coprime and that for all
the set generates .
We show that if is coprimely-invariably generated, then can be
generated with three elements, or two if is soluble, and that has zero
presentation rank. As a corollary, we show that if is any finite group such
that no proper subgroup has the same exponent as , then has zero
presentation rank. Furthermore, we show that every finite simple group is
coprimely-invariably generated.
Along the way, we show that for each finite simple group , and for each
partition of the primes dividing , the product of the
number of conjugacy classes of -elements satisfies
$\prod_{i=1}^u k_{\pi_i}(S) \leq \frac{|S|}{2| Out S|}.
Recommended from our members
Computing with symmetries
Group theory is the study of symmetry, and has many applications both within and outside mathematics. In this snapshot, we give a brief introduction to symmetries, and how to compute with them
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
A note on the probability of generating alternating or symmetric groups
We improve on recent estimates for the probability of generating the
alternating and symmetric groups and . In
particular we find the sharp lower bound, if the probability is given by a
quadratic in . This leads to improved bounds on the largest number
such that a direct product of copies
of can be generated by two elements
Using Graduation Rates to Develop Recruitment Strategies at Purdue University
This is the published version of the book chapter, made available with the permission of The American Association of Collegiate Registrars and Admissions Officers. The entire volume is available from the publisher: http://www4.aacrao.org/publications/catalog.php?category=13
Irredundant bases for the symmetric group
An irredundant base of a group acting faithfully on a finite set
is a sequence of points in that produces a strictly descending chain
of pointwise stabiliser subgroups in , terminating at the trivial subgroup.
Suppose that is or acting
primitively on , and that the point stabiliser is primitive in its
natural action on points. We prove that the maximum size of an irredundant
base of is , and in most cases . We also show that these bounds are best possible
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