Subgroup Embeddings in the Symmetric Group of Degree Nine

This paper is devoted to the determination of embedding properties of all nonprimary subgroups of the symmetric group on nine letters and continues previous investigations of the authors. The above-mentioned subgroups were tested for the following properties: abnormality, pronormality, paranormality, their weak analogs, weak normality, and the subnormalizer condition. The technique of Burnside marks, as well as respective information on the table of marks of from the TOM library of the computer algebra package GAP (Groups, Algorithms and Programming), were used. The subgroups of prime-power orders were not considered, because many of the above-mentioned concepts coincide for such subgroups. The total number of subgroups considered turned out to be 432. Bibliography: 18 titles.     

Embedding properties of nonprimary subgroups of the symmetric group of degree eight

The paper is devoted to studying some subgroup embedding properties for 110 subgroups of the symmetric group of degree 8. We deal with those nonprimary subgroups whose degree is exactly eight. The results were obtained by using of the computer algebra system GAP. They are summarized in a table. It was found that every polynormal subgroup is also paranormal. Bibliography: 12titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 124–128.     

Conjugacy in Permutation Representations of the Symmetric Group

Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e., that conjugacy classes of S _n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S _n  × S _n . We also consider the permutation representations of S _n which arise from the action of S _n on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not.     

THE FISCHER-CLIFFORD MATRICES OF THE GROUP 26: SP 6(2)

Abstract

The smallest Fischer sporadic simple group F₂₂ has 14 maximal subgroups up to conjugacy as listed in [2] and [15]. The group 2⁶: SP ₆(2) is a maximal subgroup of F ₂₂ of index 694980. We use the technique of the Fischer-Clifford matrices to constuct the character table of the group 2⁶: SP ₆(2). The Fischer-Clifford matrices of 2⁶: SP ₆(2) are used, together with the character tables of SP ₆(2) and a maximal subgroup 2⁵:S ₆ SP ₆ (2) of index 63, to construct the character table of 2⁶: SP ₆ (2).     

Conjugacy Classes,Class Sums and Character Tables for Hopf Algebras

We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras and show that the conjugacy classes are obtained from the factorization of H as irreducible left D(H)-modules. For quasitriangular semisimple Hopf algebras H, we prove that the product of two class sums is an integral combination of the class sums up to d ^?2 where d = dim H. We show also that in this case the character table is obtained from the S-matrix associated to D(H). Finally, we calculate explicitly the generalized character table of D(kS ₃), which is not a character table for any group. It moreover provides an example of a product of two class sums which is not an integral combination of class sums.     

Theorem about the conjugacy representation ofS n

Every group has two natural representations on itself, the regular representation and the conjugacy representation. We know everything about the construction of the regular representation, but we know very little about the conjugacy representation (for uncommutative groups). In this paper we will see that every irreducible complex character ofS _n (n>2) is a constituent of conjugacy character ofS _n .     

Using Mathematica to compute conjugacy classes

The conjugacy classes of finite groups play an important role in the representation theory of those groups, and it is useful to be able to compute the conjugacy classes quickly. A procedure is developed and then implemented with Mathematica to discover these conjugacy classes. The computations make use of the Cayley table in its regular form for the group. The conjugacy classes for C_4v, the point symmetry group of the square, are displayed.     

TheS 3-conjecture for solvable groups

This paper solves theS ₃-conjecture for solvable groups proving that a nontrivial finite solvable group in which no two distinct conjugacy classes have the same order is isomorphic toS ₃.     

Two maximal subgroups of E8(2)

We show that E₈(2) has a unique conjugacy class of subgroups isomorphic to PSp₄(5) and a unique conjugacy class of subgroups isomorphic to PSL₃(5). There normalizers are maximal subgroups of E₈(2) and are, respectively, isomorphic to PGSp₄(5) and Aut(PSL₃(5)).     

Bounds in the Theory of Finite Covers

We give an upper bound for the number of conjugacy classes of closed subgroups of the full wreath product FWr_WSym(Ω) which project onto Sym(Ω). Here, Ω is infinite, W is the set of n-tuples of distinct elements from Ω (for some finite n), F is a finite nilpotent group, and the topology on the wreath product is that of pointwise convergence in its imprimitive permutation action. The result addresses a problem which arises in a natural model-theoretic context about classifying certain types of finite covers.     

Groups with many subgroups having a transitive normality relation

A group is said to be aT-group if all its subnormal subgroups are normal. The structure of groups satisfying the minimal condition on subgroups that do not have the propertyT is investigated. Moreover, locally soluble groups with finitely many conjugacy classes of subgroups which are notT-groups are characterized.     

On intersections of conjugacy classes and bruhat cells

For a connected semisimple algebraic group G over an algebraically closed field k and a fixed pair (B, B ^– ) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB ^– is nonempty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution m _C ∈ 2 W associated to C. We prove that the element m _C is the unique maximal length element in its conjugacy class in W, and we classify all such elements in W. For G = SL(n + 1; k), we describe m _C explicitly for every conjugacy class C, and when w ∈ W ≌ S_n+1 is an involution, we give an explicit answer to when C ∩ (BwB) is nonempty.     

Embeddings of the groupL(2,13) in groups of lie typeE 6

In this paper we show there is exactly one conjugacy class of subgroups ofE ₆(ℂ) isomorphic toL(2, 13) with each of the characters 13+14 and 1+12+14 on a 27-dimensional module forE ₆. The one with character 13+14 is a subgroup of the irreducible closed subgroup of typeG ₂. There is a unique conjugacy class for each of the three algebraic conjugate characters 1+12+14. Our arguments have applications to fields of characteristic prime to |L(2, 13)|. Dedicated to John Thompson for his keen interest in broad areas of mathematics and in mathematicians     

Spin Representations,Powers of 2 and the Glaisher Map

Generalizing results on spin character degrees, we determine for a given conjugacy class of odd type in the double cover of S_n spin characters of S_n which have the minimal 2-power on this class in their character value. Surprisingly, the Glaisher map plays an important rôle here. Presented by L. Le BruynMathematics Subject Classifications (2000) 20C30, 05A15.     

Constructing some designs invariant under PSL2(q), q even

In this paper, we use Key-Moori methods 1 and 2 to construct some designs from the maximal subgroups and conjugacy classes of the group PSL₂(q), where q is a power of 2.     

On Complementability of Nonmetacyclic Subgroups in Finite A-Groups

We study groups G that satisfy the following conditions: (i) G is a finite solvable group with nonprimary metacyclic second subgroup and (ii) all Sylow subgroups of the group G are elementary Abelian subgroups. We describe the structure of groups of this type with complementable nonmetacyclic subgroups.     

On the unsolvability of the conjugacy problem for subgroups of the groupR 5 of pure braids

The main objective of the paper is the proof of the unsolvability of the conjugacy problem for subgroups of the pure braid groupR ₅. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 15–22, January, 1999.     

The conjugacy character ofS n tends to be regular

Consider the regular and the conjugacy characters ofS _n as vectors in Euclidean space, with the standard inner product. Asn grows, the angle between them tends to zero and the ratio of their lengths tends to one. The two characters have therefore asymptotically similar decompositions into irreducible components.     

Asymptotics of characters of symmetric groups, genus expansion and free probability

The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys-Murphy element involves many conjugacy classes with complicated coefficients. In this article, we consider a combinatorial setup which allows us to manipulate such products easily: to each conjugacy class we associate a two-dimensional surface and the asymptotic properties of the conjugacy class depend only on the genus of the resulting surface. This construction closely resembles the genus expansion from the random matrix theory. As the main application we study irreducible representations of symmetric groups S_q for large q. We find the asymptotic behavior of characters when the corresponding Young diagram rescaled by a factor q^-1/2 converge to a prescribed shape. The character formula (known as the Kerov polynomial) can be viewed as a power series, the terms of which correspond to two-dimensional surfaces with prescribed genus and we compute explicitly the first two terms, thus we prove a conjecture of Biane.     

Primitive permutation groups of simple diagonal type

LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT ^m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT ^m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic. GivenT ^m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal subgroups are abstractly isomorphic toT ^m. Secondly, form the direct productS _m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS _m×OutT whose projections inS _m are primitive. The second result of the paper is that there is a bijection between these two sets. The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which can fall into a single abstract isomorphism class.