Search

We examine the family of generalized Laguerre polynomials L n (n) (x) . In 1989, Gow discovered that if n is even, then the discriminant of L n (n) (x) is a nonzero square of a rational number. Additionally, in the case where the polynomial L n (n) (x) is irreducible over the rationals, the associated Galois group is the alternating group A n . Filaseta et al. (2012) established the irreducibility of L n (n) (x) for every n > 2 satisfying 2 (mod 4) . They also demonstrated that if n is 0 (mod 4) , then L n (n) (x) has a linear factor if it is not irreducible. The question of whether L n (n) (x) has a linear factor when n is 0 (mod 4) remained unanswered. We resolve this question by proving that L n (n) (x) does not have a linear factor for sufficiently large n. This conclusion completes the classification of generalized Laguerre polynomials having Galois group the alternating group, excluding a finite set of exceptions. [ABSTRACT FROM AUTHOR]

Let D n , k be the set of all permutations of the symmetric group S n that have no cycles of length i for all 1 ≤ i ≤ k . In the paper mentioned above, Ku, Lau, and Wong prove that the set of all the largest independent sets of the Cayley graph Cay (S n , D n , k) is equal to the set of all the largest independent sets in the derangement graph Cay (S n , D n , 1) , provided n is sufficiently large in terms of k. We give a simpler proof that holds for all n, k and also applies to the alternating group. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the classification of block-transitive 3-designs. Let D = (P , B) be a nontrivial 3- (v , 4 , λ) design and G ≤ A u t (D) acts block-transitively on D with sporadic or alternating socle, then there are exactly 6 incomplete designs as follows: D is isomorphic to a 3- (12 , 4 , λ) design with λ ∈ { 3 , 6 } , and G ≅ M 11 ; D is isomorphic to a 3- (22 , 4 , λ) design with λ ∈ { 3 , 16 } , and S o c (G) = M 22 ; D is isomorphic to a 3-(10, 4, 1) design, and G ≅ M 10 , P G L 2 (9) or P Γ L 2 (9) ; D is isomorphic to a 3-(10, 4, 6) design, and S o c (G) = A 6 . [ABSTRACT FROM AUTHOR]

Recall that a subgroup A of a group G is called G-permutable in G if for every subgroup B of G there exists an element x ∈ G such that A and B x commute. The following question was posed in the Kourovka Notebook: is there an integer n such that for all m > n the alternating group A m has no non-trivial A m -permutable subgroups? We give a positive answer to this question. Moreover, in the case of prime p we prove that A p has no non-trivial A p -permutable subgroups except p = 5 . [ABSTRACT FROM AUTHOR]

This paper contributes to the classification of flag-transitive 2-designs. Let D be a 2- (v , k , λ) design with prime square replication number. Suppose that G is a flag-transitive, point-primitive automorphism group of D with socle an alternating group. We prove that D is the unique 2-(10, 6, 5) design with G = A l t (6) or Sym(6). [ABSTRACT FROM AUTHOR]

Let C 2 ≀ H n denote the wreath product of a cyclic group C 2 of order 2 with a subgroup H n of the symmetric group S n on n letters, and let A be the direct product of a cyclic group of order 2 u with u ≥ 1 and a cyclic group of order 2 v with u ≥ v ≥ 0 . The number of homomorphisms from A to C 2 ≀ H n is denoted by h (A , C 2 ≀ H n) . Let A n be the alternating group on n letters. When H n = A n , h (A , C 2 ≀ A n) has some 2-adic properties which are similar to those of h (A , C 2 ≀ S n) . The exponent of 2 in the decomposition of h (A , C 2 ≀ H n) into prime factors is denoted by ord2 (h (A , C 2 ≀ H n)) . Let [ x ] denote the largest integer not exceeding a real number x . For each n , τ ¯ 2 (u , v) (n) denotes ∑ j = 0 u - 1 [ n / 2 j ] + [ n / 2 u + 1 ] - [ n / 2 u + 2 ] if u = v and denotes ∑ j = 0 u - 1 [ n / 2 j ] - (u - v) [ n / 2 u ] if u ≥ v + 1 , which is known to be the lower bound of ord2 (h (A , C 2 ≀ S n)) . There are three types of 2-adic properties of h (A , C 2 ≀ A n) . The lower bound of ord2 (h (A , C 2 ≀ A n)) is τ ¯ 2 (u , v) (n) if u + δ v 0 ≤ v + 2 , and is τ ¯ 2 (u , v) (n) - 1 if u + δ v 0 ≥ v + 3 . Suppose that u + δ v 0 ≥ v + 2 . For any positive odd integer y , ord2 (h (A , C 2 ≀ A 2 u y)) and ord2 (h (A , C 2 ≀ A 2 u y + 1)) are described by certain 2-adic integers. The values { h (A , C 2 ≀ A n) } n = 0 ∞ are explained by certain 2-adic analytic functions. The results are obtained by using an explicit description of the generating function ∑ n = 0 ∞ h (A , C 2 ≀ A n) X n / (2 n n !) . [ABSTRACT FROM AUTHOR]

The simple symplectic group Sp(8, 2) has 11 conugacy classes of maximal subgroups. The fourth maximal subgroup of Sp(8, 2) is a group of the form 2 10 : A 8 : = G ¯. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford–Fischer Theory. We determined the inertia factor groups of G ¯ and there are 7 such groups having the forms: H 1 = A 8 , H 2 = 2 3 : G L (3 , 2) , H 3 = 2 4 : (S 3 × S 3) , H 4 = 2 3 : S 4 , H 5 = S 5 , H 6 = (S 3 × S 3) : 2 and H 7 = 2 × S 4. The character table of G ¯ is a 81 × 81 complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 16. [ABSTRACT FROM AUTHOR]

Let be a class of finite groups containing a group of even order and closed under subgroups, homomorphic images, and extensions. Then each finite group possesses a maximal -subgroup of odd index and the study of the subgroups can be reduced to the study of the so-called submaximal -subgroups of odd index in simple groups. We prove a theorem that deduces the description of submaximal -subgroups of odd index in an alternating group from the description of maximal -subgroups of odd index in the corresponding symmetric group. In consequence, we classify the submaximal soluble subgroups of odd index in alternating groups up to conjugacy. [ABSTRACT FROM AUTHOR]

Let A be the direct product of a cyclic group of order 2 u with u ≥ 1 and a cyclic group of order 2 v with u ≥ v ≥ 0 . There are some 2-adic properties of the number h (A , A n) of homomorphisms from A to the alternating group A n on n-letters, which are similar to those of the number of homomorphisms from A to the symmetric group on n-letters. The exponent of 2 in the decomposition of h (A , A n) into prime factors is denoted by ord 2 (h (A , A n)) . Let [x] denote the largest integer not exceeding a real number x. For any nonnegative integer n, the lower bound of ord 2 (h (A , A n)) is ∑ j = 1 u [ n / 2 j ] + [ n / 2 u + 2 ] - [ n / 2 u + 3 ] - 1 if u = v ≥ 1 , and is ∑ j = 1 u [ n / 2 j ] - (u - v) [ n / 2 u + 1 ] - 1 otherwise. For any positive odd integer y, ord 2 (h (A , A 2 u + 1 y)) and ord 2 (h (A , A 2 u + 1 y + 1)) are described by certain 2-adic integers if either u ≥ v + 2 ≥ 3 or u ≥ 1 and v = 0 . The values { h (A , A n) } n = 0 ∞ are explained by certain 2-adic analytic functions unless u = v + 1 ≥ 2 . The results are obtained by using the generating function ∑ n = 0 ∞ h (A , A n) X n / n ! . [ABSTRACT FROM AUTHOR]

Let G be the alternating group Alt (n) on n letters. We prove that for any ε > 0 , there exists N = N (ε) ∈ N such that whenever n ≥ N and A, B, C, D are normal subsets of G each of size at least | G | 1 / 2 + ε , then A B C D = G . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the properties of a random walk on the alternating group A n generated by three cycles of the form (i , n - 1 , n) and (i , n , n - 1) . We call this the transpose top-2 with random shuffle. We find the spectrum of the transition matrix of this shuffle. We show that the mixing time is of order n - 3 2 log n and prove that there is a total variation cutoff for this shuffle. [ABSTRACT FROM AUTHOR]

For each positive integer n, we introduce a monoidal category J P (n) using a generalization of partition diagrams. When the characteristic of the ground field is either 0 or at least n, we show J P (n) is monoidally equivalent to the full subcategory of Rep(An) whose objects are tensor powers of the natural n-dimensional permutation representation of the alternating group An. [ABSTRACT FROM AUTHOR]

Let G be a finite group and X be a conjugacy class of G. The rank of X in G, denoted by rank(G : X), is defined to be the minimal number of elements of X generating G. In this paper we establish some general results on the ranks of certain conjugacy classes of elements for simple alternating group A n . We apply these general results together with the structure constants method to determine the ranks of all the non-trivial classes of A 8 and A 9 . [ABSTRACT FROM AUTHOR]

We develop an approach to constructing and classifying semifield projective planes with the use of a spread set. The famous conjecture is discussed on the solvability of the full collineation group of a finite semifield nondesarguesian plane. We construct a matrix representation of a spread set of a semifield plane of odd order admitting an autotopism subgroup isomorphic to the alternating group A5 and find a series of semifield planes of odd order not admitting A5. [ABSTRACT FROM AUTHOR]

We develop an approach to constructing and classifying semifield projective planes with the use of a spread set. The famous conjecture is discussed on the solvability of the full collineation group of a finite semifield nondesarguesian plane. We construct a matrix representation of a spread set of a semifield plane of odd order admitting an autotopism subgroup isomorphic to the alternating group A5 and find a series of semifield planes of odd order not admitting A5. [ABSTRACT FROM AUTHOR]

Young's orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a symmetric group. In this article, we describe a recursive algorithm to write down the expansion of each Gelfand-Tsetlin basis vector for an irreducible representation of an alternating group in terms of Young's orthogonal basis of the ambient representation of the symmetric group. This algorithm is implemented with the Sage Mathematical Software. [ABSTRACT FROM AUTHOR]

This paper is dealing with two split extensions of the form $$2^{8}{:}A_{9}.$$ We refer to these two groups by $$\overline{G}_{1}$$ and $$\overline{G}_{2}.$$ For $$\overline{G}_{1},$$ the 8-dimensional GF(2)-module is in fact the deleted permutation module for $$A_{9}.$$ We firstly determine the conjugacy classes of $$\overline{G}_{1}$$ and $$\overline{G}_{2}$$ using the coset analysis technique. The structures of inertia factor groups were determined for the two extensions. The inertia factor groups of $$\overline{G}_{1}$$ are $$A_{9},\,A_{8},\, S_{7},\,(A_{6} \times 3){:}2 $$ and $$(A_{5} \times A_{4}){:}2,$$ while the inertia factor groups of $$\overline{G}_{2}$$ are $$A_{9},\, PSL(2,8){:}3$$ and $$2^{3}{:}GL(3,2).$$ We then determine the Fischer matrices for these two groups and apply the Clifford-Fischer theory to compute the ordinary character tables of $$\overline{G}_{1}$$ and $$\overline{G}_{2}.$$ The Fischer matrices of $$\overline{G}_{1}$$ and $$\overline{G}_{2}$$ are all integer valued, with sizes ranging from 1 to 9 and from 1 to 4 respectively. The full character tables of $$\overline{G}_{1}$$ and $$\overline{G}_{2}$$ are $$84 \times 84$$ and $$40 \times 40$$ complex valued matrices respectively. [ABSTRACT FROM AUTHOR]

The partition algebra $$\mathsf {P}_k(n)$$ and the symmetric group $$\mathsf {S}_n$$ are in Schur-Weyl duality on the k-fold tensor power $$\mathsf {M}_n^{\otimes k}$$ of the permutation module $$\mathsf {M}_n$$ of $$\mathsf {S}_n$$ , so there is a surjection $$\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})$$ , which is an isomorphism when $$n \ge 2k$$ . We prove a dimension formula for the irreducible modules of the centralizer algebra $$\mathsf {Z}_k(n)$$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities of the irreducible $$\mathsf {S}_n$$ -modules in $$\mathsf {M}_n^{\otimes k}$$ . Our dimension expressions hold for any $$n \ge 1$$ and $$k\ge 0$$ . Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on $$\mathsf {M}_n^{\otimes k}$$ and the quasi-partition algebra corresponding to tensor powers of the reflection representation of $$\mathsf {S}_n$$ . [ABSTRACT FROM AUTHOR]

We develop an approach to constructing and classification of semifield projective planes with the use of a linear space and a spread set. We construct a matrix representation of the spread set of a semifield plane of odd order that admits a Baer involution in the translation complement or a subgroup of autotopisms isomorphic to the alternating group A . We give examples of semifield planes of order 81 satisfying the above indicated condition. [ABSTRACT FROM AUTHOR]

Let G = A , a finite alternating group. We study the commuting graph of G and establish, for all possible values of n barring 13, 14, 17, and 19 whether or not the independence number is equal to the clique-covering number. [ABSTRACT FROM AUTHOR]

Let G be a finite group G, and let N(G) be the set of sizes of its conjugacy classes. It is shown that, if N(G) equals N(Alt) or N(Sym), where n > 1361, then G has a composition factor isomorphic to an alternating group Altm with m ≤ n and the interval ( m, n] contains no primes. [ABSTRACT FROM AUTHOR]

For G a finite group, κ(G) is the probability that σ, τ ∈ G are conjugate, when σ and τ are chosen independently and uniformly at random. Recently, Blackburn et al. (J Lond Math Soc 86(2):755-778, 2012) gave an elementary proof that κ (Sn) ~ A/n² as n → ∞ for some constant A--a result which was first proved by Flajolet et al. (Electron J Comb 13(1):35, 2006). In this paper, we extend the elementary methods of Blackburn et al. to show that κ(An) ~ B/n² as n → ∞ for some constant B, given explicitly in this paper. [ABSTRACT FROM AUTHOR]

Consider the flag-transitive 2-( v, k, λ) symmetric designs with ( k, λ) = 1.We prove that if D is a nontrivial 2-( v, k, λ) symmetric design with ( k, λ) = 1 and G ≤ Aut( D) is flag-transitive with Soc( G) = A for n ≥ 5, then D is the projective space PG(3, 2) and G = A. [ABSTRACT FROM AUTHOR]

We improve on recent estimates for the probability of generating the alternating and symmetric groups A and S. In particular, we find the sharp lower bound if the probability is given by a quadratic in n. This leads to improved bounds on the largest number h(A) such that a direct product of h(A) copies of A can be generated by two elements. [ABSTRACT FROM AUTHOR]

There are some graphs related to the structure of a finite group, e.g. the prime graph, the solvable graph, and the noncommuting graph of a finite group. In this paper, we consider characterization by order and prime graph for alternating groups. In particular, we show that there exists a closed relation between characterization by order and prime graph of some of the alternating groups and Goldbach's conjecture. Also by using the main results, we answer some conjectures and a problem about characterization by order and degree pattern for alternating and symmetric groups that are stated in [R. Kogani-Moghaddam and A. R. Moghaddamfar, 'Groups with the same order and degree pattern,' Sci. China Math., 2012]. [ABSTRACT FROM AUTHOR]

We prove that the ranks of central unit groups of integral group rings of alternating groups of degrees greater than 38 are at least 11. The presented tables contain the ranks of all central unit groups of integral group rings of alternating groups of degrees at most 200. In particular, for every r ∈ {0, ..., 10}, we obtain the complete list of integers n such that the central unit group of the integral group ring of the alternating group of degree n has rank r. [ABSTRACT FROM AUTHOR]

It is proved that, in a nonsolvable finite symmetric or alternating group except for the group S, for any pair of nilpotent subgroups there exists a subgroup conjugate to one of them such that its intersection with the other subgroup is trivial. [ABSTRACT FROM AUTHOR]

The spectrum of a group is the set of its element orders. A finite group G is said to be recognizable by spectrum if every finite group that has the same spectrum as G is isomorphic to G. It is proved that simple alternating groups A are recognizable by spectrum, for n ≠ 6, 10. This implies that every finite group whose spectrum coincides with that of a finite non-Abelian simple group has at most one non-Abelian composition factor. [ABSTRACT FROM AUTHOR]

All instances of coincidence between the prime graphs of nonabelian simple groups G and S are found, where G is an alternating group of degree n ≥ 5 and S is a nonabelian finite simple group. The precise bound of the maximal number of pairwise nonisomorphic nonabelian simple groups with the same prime graph is given in the case that one of these groups is an alternating group. [ABSTRACT FROM AUTHOR]

We give explicit, asymptotically sharp bounds for the probability that a pair of random permutations of degree n generates either S or A and also for the probability that a pair of random even permutations of degree n generates A. As an application we answer a question of Wiegold in the case of alternating groups. [ABSTRACT FROM AUTHOR]

Let $${\mathcal{D}}$$ be a nontrivial triplane, and G be a subgroup of the full automorphism group of $${\mathcal{D}}$$. In this paper we prove that if $${\mathcal{D}}$$ is a triplane, $${G\leq Aut(\mathcal{D})}$$ is flag-transitive, point-primitive and Soc( G) is an alternating group, then $${\mathcal{D}}$$ is the projective space PG2(3, 2), and $${G\cong A_7}$$ with the point stabiliser $${G_x\cong PSL_3(2)}$$. [ABSTRACT FROM AUTHOR]

The spectrum of a finite group is the set of its element orders. We describe the composition structure of every finite group with the same spectrum as that of the alternating group of degree 10 and not isomorphic to it. This group is isomorphic to the semidirect product of the abelian {3, 7}-group, which contains an element of order 21, by the symmetric group of degree 5. [ABSTRACT FROM AUTHOR]

In this note, we construct a 2-basic set of the alternating group $${\mathfrak{A}_n}$$. To do this, we construct a 2-basic set of the symmetric group $${\mathfrak{S}_n}$$ with an additional property, such that its restriction to $${\mathfrak{A}_n}$$ is a 2-basic set. We adapt here a method developed by Brunat and Gramain (J. Reine Angew. Math., to appear) for the case when the characteristic is odd. One of the main tools is the generalized perfect isometries defined by Külshammer et al. (Invent. Math. 151, 513–552, (2003)). [ABSTRACT FROM AUTHOR]

A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two sufficient and necessary conditions for such graphs to be 1- or 2-arcregular are given and based on the conditions, several infinite families of 1- or 2-arc-regular cubic Cayley graphs of alternating groups are constructed. [ABSTRACT FROM AUTHOR]

Previously, we dubbed the conjecture that the alternating group An has no semiproportional irreducible characters for any natural n [1]. This conjecture was then shown to be equivalent to the following [3]. Let α and β be partitions of a number n such that their corresponding characters χα and χβ in the group Sn are semiproportional on An. Then one of the partitions α or β is self-associated. Here, we describe all pairs (α, β) of partitions satisfying the hypothesis and the conclusion of the latter conjecture. [ABSTRACT FROM AUTHOR]

Previously, we dubbed the conjecture that the alternating group An has no semiproportional irreducible characters for any natural n [1]. This conjecture was then shown to be equivalent to the following [3]. Let α and β be partitions of a number n such that their corresponding characters χα and χβ in the group Sn are semiproportional on An. Then one of the partitions α or β is self-associated. Here, we describe all pairs (α, β) of partitions satisfying the hypothesis and the conclusion of the latter conjecture. [ABSTRACT FROM AUTHOR]

Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every two non-commuting members of X generate a subgroup isomorphic to A4 or to A5. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they generate a subgroup isomorphic to A4. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X) and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A5 then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2) The subgroup 〈XG〉 is a direct product of subgroups 〈 C α〉-generated by some connected components C α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆ G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup isomorphic to A5. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A5 or is cyclic of order 5. [ABSTRACT FROM AUTHOR]

Let P(n) be the set of all partitions of a natural number n. In the representation theory of symmetric groups, for every partition α ∈ P(n), the partition h(α) ∈ P(n) is defined so as to produce a certain set of zeros in the character table for Sn. Previously, the analog f(α) of h(α) was obtained pointing out an extra set of zeros in the table mentioned. Namely, h(α) is greatest (under the lexicographic ordering ≤) of the partitions β of n such that χα(gβ) ≠ 0, and f(α) is greatest of the partitions γ of n that are opposite in sign to h(α) and are such that χα(gγ) ≠ 0, where χα is an irreducible character of Sn, indexed by α, and gβ is an element in the conjugacy class of Sn, indexed by β. For α ∈ P(n), under some natural restrictions, here, we construct new partitions h′(α) and f′(α) of n possessing the following properties. (A) Let α ∈ P(n) and n ⩾ 3. Then h′(α) is identical is sign to h(α), χα(gh′(α)) ≠ 0, but χα(gγ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of h(α), and h′(α) < γ < h(α). (B) Let α ∈ P(n), α ≠ α′, and n ⩾ 4. Then f′(α) is identical in sign to f(α), χα(gf′(α)) ≠ 0, but χα(gγ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of f(α), and f′(α) < γ < f(α). The results obtained are then applied to study pairs of semiproportional irreducible characters in An. [ABSTRACT FROM AUTHOR]

It is proved that a group G generated by a conjugacy class X of elements of order 3, so that every two non-commuting elements of X generate a subgroup isomorphic to an alternating group of degree 4 or 5, is locally finite. More precisely, either G contains a normal elementary 2-subgroup of index 3, or G is isomorphic to an alternating group of permutations on some (possibly infinite) set. [ABSTRACT FROM AUTHOR]

It is proved that values of all complex characters for a Sylow 2-subgroup P of any Weyl group are rational, and every element of P is a product of two involutions in P. Similar results hold also for Sylow 2-subgroups of alternating groups. [ABSTRACT FROM AUTHOR]

On Aschbacher's definition, a subgroup $N$ of a finite group $G$ is called a $p$-superlocal for a prime $p$ if $N=N\_G(O\_p(N))$. We describe the $p$-superlocals in symmetric and alternating groups, thereby resolving part way Problem 11.3 in the Kourovka Notebook [3]. [ABSTRACT FROM AUTHOR]

Groups with complemented subgroups, which are also called completely factorizable groups, were studied by P. Hall, S. N. Chernikov, arid N. V. Chernikova (Baeva). For complete factorizability, it is sufficient (Theorem 1) that each proper subgroup have a normal complement in some larger subgroup. A group is said to be weakly factorizable if each of its proper subgroups is complemented in some larger subgroup; the problem of describing finite groups with this property is posed (Question 8.31) in the "Kourovka Notebook." Some properties of these groups are considered. The question is studied for Sylow p-subgroups of Chevalley-type groups of characteristic p. The main theorem, Theorem 2, establishes the weak factorizability of the Sylow p -subgroups in the symmetric arid alternative groups arid in the classical linear groups over fields of characteristic p > 0, excluding the unitary groups of odd dimension > p. [ABSTRACT FROM AUTHOR]

A finite group G is a Dπ-group for some set π of primes if maximal π-subgroups of G are all conjugate. Assume that every non-Abelian composition factor of the Dπ-group G is isomorphic either to an alternating group, or to one of the sporadic groups, or to a simple group of Lie type over a field whose characteristic belongs to π. We prove that an extension of G by an arbitrary Dπ-group and every normal subgroup of G are Dπ-groups. This gives partial answers to Questions 3.62 and 13.33 in the “Kourovka Notebook.” Also, we describe all Dπ-groups whose composition factors are isomorphic to alternating, sporadic, and Lie-type groups whose characteristics belong to π. And bring to a close the description of Hall subgroups in sporadic groups, initiated by F. Gross. [ABSTRACT FROM AUTHOR]

The paper gives a description the permutations from the alternating group A n that, for a given positive integer k≥4, cannot be presented as a product of two permutations each of which contains only cycles of lengths 1 and 4 in the expansion into independent cycles. We construct a set Q k such that, for each n from Q k, the group A n contains a permutation not representable in the above form. We give answers to two questions of Brenner and Evans on the representability of even permutations in the form of a product of two permutations of a given order k. [ABSTRACT FROM AUTHOR]

A classification of maximal subgroups of odd index in finite groups with alternating socle is obtained. [ABSTRACT FROM AUTHOR]

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. 'Classroom' is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. [ABSTRACT FROM AUTHOR]

We prove that alternating groups with three prime graph components are uniquely determined by the set of orders of maximal abelian subgroups. [ABSTRACT FROM AUTHOR]