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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]

We study the height of generators of Galois extensions of the rationals having the alternating group A n as Galois group. We prove that if such generators are obtained from certain, albeit classical, constructions, their height tends to infinity as n increases. This provides an analogue of a result by Amoroso, originally established for the symmetric group. [ABSTRACT FROM AUTHOR]

In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups $G$ are studied in respect of formation of lattices ${\rm L}(G)$ and sublattices of ${\rm L}(G)$. It is proved that the collections of all pronormal subgroups of ${\rm A}_n$ and S$_n$ do not form sublattices of respective ${\rm L}({\rm A}_n)$ and ${\rm L}({\rm S}_n)$, whereas the collection of all pronormal subgroups ${\rm LPrN}({\rm Dic}_n)$ of a dicyclic group is a sublattice of ${\rm L}({\rm Dic}_n)$. Furthermore, it is shown that ${\rm L}({\rm Dic}_n)$ and ${\rm LPrN}({\rm Dic}_n$) are lower semimodular lattices.

We consider representations of an arbitrary permutation π of degree 2n, n ⩾ 3, by products of the so-called (2n)-permutations (any cycle of such a permutation has length 2). We show that any even permutation is represented by the product of four (2n)-permutations. Products of three (2n)-permutations cannot represent all even permutations. Any odd permutation is realized (for odd n) by a product of five (2n)-permutations. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the classification of all pairs (T,G) $({\mathscr{T}},G)$, where T ${\mathscr{T}}$ is a triple system and G $G$ is a block‐transitive but not flag‐transitive automorphism group of T ${\mathscr{T}}$. We prove that if the socle of G $G$ is a sporadic or alternating group, then one of the following holds: (i)T ${\mathscr{T}}$ is a TS(10,2) $TS(10,2)$ and G≅A5 $G\cong {A}_{5}$;(ii)T ${\mathscr{T}}$ is a TS(10,4) $TS(10,4)$ and G≅S5 $G\cong {S}_{5}$;(iii)T ${\mathscr{T}}$ is a TS(55,28) $TS(55,28)$ and G≅A11 $G\cong {A}_{11}$ or S11 ${S}_{11}$;(iv)T ${\mathscr{T}}$ is a TS(55,λ) $TS(55,\lambda)$ with λ∈{4,8,16} $\lambda \in \{4,8,16\}$ and G≅M11 $G\cong {M}_{11}$. [ABSTRACT FROM AUTHOR]

Let G be a finite group and Irr(G) the set of all irreducible complex characters of G. Define the codegree of X ∈ Irr(G) as cod(x) := |G : ker(x)|/x(1) and let cod(G) := {cod(x) | x ∈ Irr(G)} be the codegree set of G. Let An be an alternating group of degree n ≥ 5. We show that An is determined up to isomorphism by cod(An). [ABSTRACT FROM AUTHOR]

We provide three generalizations of all best known generating sets for S n and A n : (1) m -cycles with consecutive entries; (2) m -cycles with fixed first ℓ entries; (3) two cycles, either (1 , ... , m) , (1 , ... , n) or (1 , ... , m) , (2 , ... , n). Our work on (3) fills a gap in the history of the problem, going back to a 1939 paper by S. Piccard [Sur les bases du groupe symétrique et du groupe alternant, Math. Ann. 116(1) (1939) 752–767]. [ABSTRACT FROM AUTHOR]

In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups G are studied in respect of formation of lattices L(G) and sublattices of L(G). It is proved that the collections of all pronormal subgroups of An and Sn do not form sublattices of respective L(An) and L(Sn), whereas the collection of all pronormal subgroups LPrN(Dicn) of a dicyclic group is a sublattice of L(Dicn). Furthermore, it is shown that L(Dicn) and LPrN(Dicn) are lower semimodular lattices. [ABSTRACT FROM AUTHOR]

Without using the classification of finite simple groups (CFSG), we show that the probability that two random elements of Sn$$ {S}_n $$ generate a primitive group smaller than An$$ {A}_n $$ is at most exp(−c(nlogn)1/2)$$ \exp \left(-c{\left(n\log n\right)}^{1/2}\right) $$. As a corollary we get Dixon's asymptotic expansion 1−1/n−1/n2−4/n3−23/n4−⋯$$ 1-1/n-1/{n}^2-4/{n}^3-23/{n}^4-\cdots $$for the probability that two random elements of Sn$$ {S}_n $$ (or An$$ {A}_n $$) generate a subgroup containing An$$ {A}_n $$. [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]

Let G be a finite group. For a given pair of non-negative integers n and k , we aim to give an explicit formula to count the number of elements of Λ k (G) , the set of all irreducible representations of G whose degree is not divisible by 2 k . In this paper, we give formulas for | Λ k (G) | , when G is symmetric group, generalized symmetric group, and alternating group. Further, we also have a complete characterization of the elements in Λ 2 (S n) , Λ 1 (G (n , r)) , and the self-conjugate partitions of 2 l and 2 l + 1 with degree not divisible by 4. [ABSTRACT FROM AUTHOR]

We use the Key-Moori Method 1 and examine 1-designs and codes from the representations of the alternating group $A_7$. It is shown that a self-dual symmetric 2-$(35,18,9)$ design and an optimal even binary $[21,14,4]$ LCD code are found such that they are invariant under the full automorphism groups $S_8$ and $S_7$, respectively. Moreover, designs with parameters 1-$(21,l,k_{1,l})$ and 1-$(35,l,k_{2,l})$ are obtained, where $\omega$ is a codeword, $l=wt(\omega)$, $k_{1,l}=l|\omega^{S_7}|/21$ and $k_{2,l}=l|\omega^{S_7}|/35$. It is seen that there exist a 2-$(21,5,12)$ design with the full automorphism group $S_7$ among these 1-designs.

We use the Key-Moori Method 1 and examine 1-designs and codes from the representations of the alternating group A7. It is shown that a self-dual symmetric 2-(35, 18, 9) design and an optimal even binary [21, 14, 4] LCD code are found such that they are invariant under the full automorphism groups S8 and S7, respectively. Moreover, designs with parameters 1-(21, l, k1,l) and 1-(35, l, k2,l) are obtained, where ω is a codeword, l = wt(ω), k1,l = l|ωS7 |/21 and k2,l = l|ωS7 |/35. It is seen that there exist a 2-(21, 5, 12) design with the full automorphism group S7 among these 1-designs. [ABSTRACT FROM AUTHOR]

This paper identifies all pairs of ordinary irreducible characters of the alternating group which agree on conjugacy classes of elements of order not divisible by a fixed integer l, for l≠3. We do likewise for spin characters of the symmetric and alternating groups. We find that the only such characters are the conjugate or associate pairs labelled by partitions with a certain parameter divisible by l. When l is prime, this implies that the rows of the l-modular decomposition matrix are distinct except for the rows labelled by these pairs. When l=3 we exhibit many additional examples of such pairs of characters. [ABSTRACT FROM AUTHOR]

Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of finitely generated groups. The best known example of a simple group is the alternating group A n , where n ≥ 5 . This article establishes a relation between the hyperbolic triangle group denoted as ▵ * (3 , 7 , r) and the alternating group. The approach involves employing coset diagrams to establish this connection. The construction of adjacency matrices for these coset diagrams is performed, followed by a detailed examination of their spectral characteristics. [ABSTRACT FROM AUTHOR]

This paper provides a complete determination of which of the alternating groups A n and the symmetric groups S n occur as the automorphism group of some regular or chiral map on an orientable surface, and which of them occur as the automorphism group of a regular map on a non-orientable surface. The situation for some given types (m , k) is also considered, where k is the valency and m is the face-size, with special focus on types with m = 3 , and more particularly with (m , k) = (3 , 7) or (3 , 8) , or their duals. Some observations are made also about what happens for regular and orientably-regular maps with given valency, and for regular and chiral polyhedra. Much but certainly not all of what is presented follows from theorems in previous papers by the author and others, and this one brings them and some new observations together into a single reference. [ABSTRACT FROM AUTHOR]

In this paper, we establish a recursive formula for the number of chains of subgroups in the subgroup lattice of the groups formed by the Cartesian products of cyclic groups of prime power order with alternating groups of degree 3. The subgroup chains were characterised by an enumerative technique derived from the set of representatives of isomorphism classes of subgroups with their sizes. [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]

A new strongly regular graph with parameters (81,30,9,12) is found as a graph invariant under certain subgroup of the full automorphism group of the previously known strongly regular graph discovered in 1981 by J. H. van Lint and A. Schrijver.

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]

The generalized Jones result in A J Kennedy et al.(2004) which says Pk(x,G); a G-vertex colored partition is a centralizer algebra of the action of the direct product of Sn, symmetric group and G (i.e., Sn x G) on the tensor products of its permutation representation. In this paper, specifically we restricted the action of the direct product of Sn, symmetric group and G (i.e., Sn × G) to the action of the direct product of an alternating group and G (i.e., An × G). Herein, we determine the basis for the centralizer algebra and exhibit that at that moment, the centralizer is isomorphic to Pk(x;G). Also, we do the same for b Pk(x;G); the Extended G-vertex colored partition algebras. [ABSTRACT FROM AUTHOR]

Counting fuzzy subgroups of a finite group is a fundamental problem of fuzzy group theory. Many researchers have made significant contributions to the rapid growth of this topic in recent years. The number of fuzzy subgroups of any group is infinite without the aid of equivalence relation. Some authors have used the equivalence relation of fuzzy sets to study the equivalence of fuzzy subgroups ([5], [6], [16]). The problem of counting the number of distinct fuzzy subgroups of a finite group is relative to the choice of the equivalence relation. The number of fuzzy subgroups of a particular group varies from one equivalence relation to the other. The equivalence relation applied in our computation can be seen in the existing literature. Sulaiman and Abd Ghafur [10] define an equivalence relation for counting fuzzy subgroups of group G. We have used this relation to find fuzzy subgroups and fuzzy normal subgroups of S2, S3 and A4. Lattice subgroup diagrams were used in our computation. [ABSTRACT FROM AUTHOR]

–Let F be a finite field of characteristic 2. In this article, we have looked into the existence of normal complement of G in V (F G) , where G is either the alternating group A4 or the dihedral group D 4 m of order 4m, for an odd integer m ≥ 3 . Also, we have explicitly found a normal complement of the symmetric group S4 in V (F S 4) over the field F containing 2 elements. [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]

Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of finitely generated groups. The best known example of a simple group is the alternating group An, where n≥5. This article establishes a relation between the hyperbolic triangle group denoted as ▵*(3,7,r) and the alternating group. The approach involves employing coset diagrams to establish this connection. The construction of adjacency matrices for these coset diagrams is performed, followed by a detailed examination of their spectral characteristics.

The prime graph (or Gruenberg-Kegel graph) of a finite group is a well-known graph. In this paper, first, we investigate the structure of the finite groups with a non-complete prime graph. Then as an application, we prove that every alternating group $A_n$, where $n\leq 31$ is determined by its order and its largest element order. Also, we show that $A_{32}$ is not characterizable by order and the largest element order.

We prove that the symmetric group Sn has a unique minimal cover M by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the size of M. In addition, we prove that the size of M is equal to the size of a maximal non-nilpotent subset of Sn. This cover M has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of n into distinct positive integers. We show that these results contrast with those for the alternating group An. In particular, we prove that, for all but finitely many values of n, no minimal cover of An by maximal nilpotent subgroups is a normal cover and the size of a minimal cover of An by maximal nilpotent subgroups is strictly greater than the size of a maximal non-nilpotent subset of An. [ABSTRACT FROM AUTHOR]

In this paper, we established the number of chains of subgroups in the subgroup lattice of the Cartesian product of the alternating group and cyclic group using computational technique induced by the set of representatives of isomorphism classes of subgroups. [ABSTRACT FROM AUTHOR]

An interesting fact is that almost all the connected 2-arc-transitive nonnormal Cayley graphs on nonabelian simple groups with small valency or prime valency (provided solvable vertex stabilizers) are Cayley graphs on alternating groups A n. This naturally motivates the study of 2-arc-transitive Cayley graphs on A n for arbitrary valency. In this paper, we characterize the automorphism groups of such graphs. In particular, we show that for a non-complete (G , 2) -arc-transitive Cayley graph on A n with G almost simple, the socle of G is either A n + 1 or A n + 2. We also construct the first infinite family of (A n + 2 , 2) -arc-transitive Cayley graphs on A n. [ABSTRACT FROM AUTHOR]

We give a classification of the generalized Laguerre polynomials L n (α) (x) , where α ≠ n is an integer, having the alternating group A n as the Galois group over the rationals for up to a finitely many exceptions. [ABSTRACT FROM AUTHOR]

A well known diameter search problem for finite groups with respect to its systems of generators is considered. The problem can be formulated as follows: find the diameter of a group over its system of generators. The diameter of a group over a specific system of generators is the diameter of the corresponding Cayley graph. It is considered alternating groups with classic irreducible system of generators consisting of cycles with length three of the form $(1,2,k)$. The main part of the paper concentrates on analysis how even permutations decompose with respect to this system of generators. The rules for moving generators from permutation's decomposition from left to right and from right to left are introduced. These rules give rise for transformations of decompositions, that do not increase their lengths. They are applied for removing fixed points of a permutation, that were included in its decomposition. Based on this rule the stability of system of generators is proved. The strict growing property of the system of generators is also proved, as the corollary of transformation rules and the stability property. It is considered homogeneous theory, that was introduced in the previous author's paper. For the series of alternating groups with systems of generators mentioned above it is shown that this series is uniform and homogeneous. It makes possible to apply the homogeneous down search algorithm to compute the diameter. This algorithm is applied and exact values of diameters for alternating groups of degree up to 43 are computed.

Let p be a prime and let G be a finite group which is generated by the set G p of its p -elements. We show that if G is solvable and not a p -group, then the minimal number σ p (G) of proper subgroups of G whose union contains G p is equal to 1 less than the minimal number of proper subgroups of G whose union is G. For p -solvable groups G , we always have σ p (G) ≥ p + 1. We study the case of alternating and symmetric groups G in detail. [ABSTRACT FROM AUTHOR]

Chiral polytopes are highly symmetric abstract polytopes that have maximal rotational symmetry but no reflections. Until recently, very few examples of chiral d -polytopes with rank d ≥ 5 were known. In this paper we extend a construction for chiral 4-polytopes (with tetrahedral facets) to produce two infinite families of chiral d -polytopes with simplex facets, for every integer d ≥ 5. Specifically, we show that for every d ≥ 5 , there exists a chiral d -polytope with automorphism group isomorphic to the alternating group A n and a chiral d -polytope with automorphism group isomorphic to the symmetric group S n , for all but finitely many n. In each case the facets are simplices, and the vertex-figures are chiral. Our extended construction can also be modified to produce four families of regular d -polytopes with simplices as facets, and automorphism groups isomorphic to A n , S n , A n × C 2 and S n × C 2 for all but finitely many n. [ABSTRACT FROM AUTHOR]

In this paper, we construct some designs invariant under the alternating groups, using a method that has been recently introduced by Moori. We consider the action of An on the 2-subsets of a set of size n. The blocks of the designs are the conjugates of the set of fixed-points of the permutation with respect to this action. For every element g ∈ A n , we give explicit formulae to compute the parameters of the designs based on the cycle representation of g. We also prove that the binary codes of these designs have minimum distances at most 4. [ABSTRACT FROM AUTHOR]

Let Sn be the symmetric group and An be the alternating group on n symbols. In this article, we have proved that if F is a finite field of characteristic p > n, then there does not exist a normal complement of Sn (n is even) and An (n ≥ 4) in their corresponding unit groups U (F S n) and U (F A n). Moreover, if F is a finite field of characteristic 3, then A4 does not have normal complement in the unit group U (F A 4). [ABSTRACT FROM AUTHOR]

Let G be a finite group which is not cyclic of prime power order. The join graph Δ(G) of G is a graph whose vertex set is the set of all proper subgroups of G, which are not contained in the Frattini subgroup G and two distinct vertices H and K are adjacent if and only if G = ⟨H;K⟩. Among other results, we show that if G is a finite cyclic group and H is a finite group such that Δ(G) ≅ Δ(H), then H is cyclic. Also we prove that Δ(G) ≅ Δ(A5) if and only if G ≅ A5. [ABSTRACT FROM AUTHOR]

Given three positive integers n , a , b with n = a b , we determine the base size of the symmetric group and of the alternating group of degree n in their action on the set of partitions into b parts having cardinality a. [ABSTRACT FROM AUTHOR]

This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours. [ABSTRACT FROM AUTHOR]

Denote the alternating and symmetric groups of degree n by A n and S n , respectively. Consider a permutation σ ∈ S n , all of whose nontrivial cycles are of the same length. We find the minimal polynomials of σ in the ordinary irreducible representations of A n and S 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 G be the alternating group of degree n. Let ω (G) be the maximal size of a subset S of G such that 〈 x , y 〉 = G whenever x , y ∈ S and x ≠ y and let σ (G) be the minimal size of a family of proper subgroups of G whose union is G. We prove that, when n varies in the family of composite numbers, σ (G) / ω (G) tends to 1 as n → ∞. Moreover, we explicitly calculate σ (A n) for n ≥ 21 congruent to 3 modulo 18. [ABSTRACT FROM AUTHOR]

In 2010, V. Futorny and S. Ovsienko gave a realization of U (gl n) as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of S 1 × S 2 × ⋯ × S n , where S j is the symmetric group on j variables. An interesting question is what a similar algebra would be in the invariant ring with respect to a product of alternating groups. In this paper we define such an algebra, denoted A (gl n) , and show that it is a Galois ring. For n = 2 , we show that it is a generalized Weyl algebra, and for n = 3 provide generators and a list of verified relations. We also discuss some techniques to construct Galois orders from Galois rings. Additionally, we study categories of finite-dimensional modules and generic Gelfand-Tsetlin modules over A (gl n). Finally, we discuss connections between the Gelfand-Kirillov Conjecture, A (gl n) , and the positive solution to Noether's problem for the alternating group. [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]