Base sizes of primitive permutation groups

Let G be a permutation group, acting on a set $$\varOmega $$ Ω of size n. A subset $${\mathcal {B}}$$ B of $$\varOmega $$ Ω is a base for G if the pointwise stabilizer $$G_{({\mathcal {B}})}$$ G ( B ) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of $$\mathrm {Sym}(n)$$ Sym ( n ) is large base if there exist integers m and $$r \ge 1$$ r ≥ 1 such that $${{\,\mathrm{Alt}\,}}(m)^r \unlhd G \le {{\,\mathrm{Sym}\,}}(m)\wr {{\,\mathrm{Sym}\,}}(r)$$ Alt ( m ) r ⊴ G ≤ Sym ( m ) ≀ Sym ( r ) , where the action of $${{\,\mathrm{Sym}\,}}(m)$$ Sym ( m ) is on k-element subsets of $$\{1,\dots ,m\}$$ { 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 $$\mathrm {M}_{24}$$ M 24 in its natural action on 24 points, or $$b(G)\le \lceil \log n\rceil +1$$ b ( G ) ≤ ⌈ log n ⌉ + 1 . Furthermore, we show that there are infinitely many primitive groups G that are not large base for which $$b(G) > \log n + 1$$ b ( G ) > log n + 1 , so our bound is optimal.

Octad Orbits for Certain Subgroups of

Using Curtis’s MOG [3], we display the orbits and orbit representatives for various subgroups of the Mathieu group acting on the octads of the Steiner system This information is deployed in [8] and [9] to study a graph associated with the largest simple Fischer group.

a×b=c in 2+1D TQFT

We study the implications of the anyon fusion equation a×b=c on global properties of 2+1D topological quantum field theories (TQFTs). Here a and b are anyons that fuse together to give a unique anyon, c. As is well known, when at least one of a and b is abelian, such equations describe aspects of the one-form symmetry of the theory. When a and b are non-abelian, the most obvious way such fusions arise is when a TQFT can be resolved into a product of TQFTs with trivial mutual braiding, and a and b lie in separate factors. More generally, we argue that the appearance of such fusions for non-abelian a and b can also be an indication of zero-form symmetries in a TQFT, of what we term "quasi-zero-form symmetries" (as in the case of discrete gauge theories based on the largest Mathieu group, M24), or of the existence of non-modular fusion subcategories. We study these ideas in a variety of TQFT settings from (twisted and untwisted) discrete gauge theories to Chern-Simons theories based on continuous gauge groups and related cosets. Along the way, we prove various useful theorems.

Hyperkähler isometries of K3 surfaces

We consider symmetries of K3 manifolds. Holomorphic symplectic automorphisms of K3 surfaces have been classified, and observed to be subgroups of the Mathieu group M23. More recently, automorphisms of K3 sigma models commuting with SU(2) × SU(2) R-symmetry have been classified by Gaberdiel, Hohenegger, and Volpato. These groups are all subgroups of the Conway group. We fill in a small gap in the literature and classify the possible hyperkähler isometry groups of K3 manifolds. There is an explicit list of 40 possible groups, all of which are realized in the moduli space. The groups are all subgroups of M23.

Postoperative complications and outcomes of Mathieu and Snodgrass techniques in a sample of Iraq patients with anterior distal shaft hypospadias

Background: Hypospadias is a common congenital anomaly of the penis in which the urethra opens proximal to its normal position at the tip of the glans. The purpose of this study was to compare the outcomes of Mathieu and Snodgrass techniques in the repair of anterior distal shaft hypospadias.Methods: From October 2009 to November 2010, forty five patients with the ages ranged 1 to 12 years suffering from anterior distal shaft hypospadias, were assessed. Inclusion criteria were anterior distal shaft hypospadias, and exclusion criteria were association with chordee, circumcision, and surgical repair history. Twenty-five patients underwent surgical repair using Snodgrass technique and 20 patients using Mathieu technique. Surgery was performed by one single surgeon, acquainted with both techniques. Patients were examined 1 week, and 1 month after discharge. Data including duration of the surgery, stenting time and any kind of complications such as break down, meatal stenosis, and fistula formation were collected. Also, success rate was calculated for every single patient and accordingly, the two groups were compared.Results: Mean operative time were 74±26minutes for Mathieu group and106.11±23 for minutes in Snodgrass group (P<0.05). Stenting mean duration was 6.8±1.1 days, in Mathieu group and, 6.3±0.8 days in Snodgrass group (P>0.05). The rate of break down, meatal stenosis, and fistula formation were 10%, 0%, and 5% in Mathieu group and 4%, 8%, and 8% in Snodgrass group respectively (P>0.05). Success rate was 88% in Snodgrass group and 85% in Mathieu group (P>0.05).Conclusions: In spite of some reports about preference for Snodgrass technique, we concluded that both techniques are as acceptable and as effective as each other for hypospodias repairing, regardless of cosmetic outcomes; however, we need further studies and larger sample sizes to determine which is the superior technique.

Finite noncrystallographic groups, 11-vertex equi-edged triangulated clusters and polymorphic transformations in metals

The one-to-one correspondence has been revealed between a set of cosets of the Mathieu groupM11, a set of blocks of the Steiner systemS(4, 5, 11) and 11-vertex equi-edged triangulated clusters. The revealed correspondence provides the structure interpretation of theS(4, 5, 11) system: mapping of the biplane 2-(11, 5, 2) onto the Steiner systemS(4, 5, 11) determines uniquely the 11-vertex tetrahedral cluster, and the automorphisms of theS(4, 5, 11) system determine uniquely transformations of the said 11-vertex tetrahedral cluster. The said transformations correspond to local reconstructions during polymorphic transformations in metals. The proposed symmetry description of polymorphic transformation in metals is consistent with experimental data.

Recognizing M10 by spectrum in the class of all groups

We prove that a group with spectrum {1, 2, 3, 4, 5, 8} is locally finite and is isomorphic to M10, where M10 = A6 ⋅ 2 is the 3-transitive Mathieu group of degree 10, i.e. the corresponding maximal subgroup of the sporadic simple group M11.