Powers of Cycle-Classes in Symmetric Groups

International Journal of Apllied Mathematics, 2013

The order of an element x in a finite group G is the smallest positive integer k, such that x k is the group identity. The set of all possible such orders joint with the number of elements that each order referred to, is called the order classes of G. In this paper, the order classes of symmetric groups is derived.

Journal of Algebra, 1998

J. Austral. Math. Soc. Ser. …, 1988

Bulletin of the Iranian Mathematical Society, 2018

By a famous result, the subgroup generated by the n-cycle σ = (1, 2,. .. , n) and the transposition τ = (a, b) is the full symmetric group S n if and only if gcd(n, b−a) = 1. In this paper, we first generalize the above result for one n-cycle and k arbitrary transpositions, and then provide similar necessary and sufficient conditions for the subgroups of S n in the following three cases: first, the subgroup generated by the ncycle σ and a 3-cycle δ = (a, b, c), second, the subgroup generated by the n-cycle σ and a set of transpositions and 3-cycles, and third, by the n-cycle σ and an involution (a, b)(c, d). In the first case, we also determine the structure of the subgroup generated by (1, 2,. .. , n) and a 3-cycle δ = (a, b, c) in general. Finally, an application to unsolvability of a certain infinite family of polynomials by radicals is given.

Let G be a finite group and let πe(G) be the set of element orders G. Let k ∈ πe(G) and let m k be the number of elements of order k in G. Set nse(G):={m k |k ∈ πe(G)}. In this paper, we prove that if G is a group such that nse(G)=nse(Sn) where n ∈ {3, 4, 5, 6, 7}, then G ∼ = Sn.

Asian-European J. Math., 2014

Let n be a positive integer, σ be an element of the symmetric group Sn and let σ be a cycle of length n. The elements α, β ∈ Sn are σ-equivalent, if there are natural numbers k and l, such that σ k α = βσ l , which is the same as the condition to exist natural numbers k 1 and l 1 , such that α = σ k 1 βσ l 1. In this work, we examine some properties of the so-defined equivalence relation. We build a finite oriented graph Γn with the help of which is described an algorithm for solving the combinatorial problem for finding the number of equivalence classes according to this relation.

Journal of the London Mathematical Society, 2001

In 1998, the second author of this paper raised the problem of classifying the irreducible characters of Sn of prime power degree. Zalesskii proposed the analogous problem for quasi-simple groups, and he has, in joint work with Malle, made substantial progress on this latter problem. With the exception of the alternating groups and their double covers, their work provides a complete solution. In this article we first classify all the irreducible characters of Sn of prime power degree (Theorem 2.4), and then we deduce the corresponding classification for the alternating groups (Theorem 5.1), thus providing the answer for one of the two remaining families in Zalesskii's problem. This classification has another application in group theory. With it, we are able to answer, for alternating groups, a question of Huppert: which simple groups G have the property that there is a prime p for which G has an irreducible character of p-power degree > 1 and all of the irreducible characters...

International journal of quantum chemistry, 1998

An algorithm for the evaluation of the structure constants in the class algebra of the symmetric group has recently been considered. The product of the class wŽ .x sum p that consists of a cycle of length p and n y p fixed points, with an arbitrary n class sum in S , was found to be expressible in terms of a set of reduced class coefficients n Ž . RCCs , the p-RCCs. The combinatorial significance of the p-RCCs is elucidated, showing that they are related to a well-defined enumeration problem within S , which has to do p with a certain refinement of the corresponding class multiplication problem. This is in contrast with the representation-theoretic evaluation of the p-RCCs, which requires the wŽ .x evaluation of products involving p for several values of n ) p. The combinatorial n interpretation of the p-RCCs allows the derivation of some of their previously conjectured properties and of some of the ''elimination rules'' that specific types of p-RCCs were found to satisfy.

Journal of Combinatorial Designs, 1999

Let m ≥ 3 be an odd integer and let k, n, and s ≥ 3 be positive integers. We present sufficient conditions for the existence of Cs-factorizations of (C m k ) n . We show these conditions to be necessary when m is prime.

Journal of Algebra, 2015

We conjecture that if G is a finite primitive group and if g is an element of G, then either the element g has a cycle of length equal to its order, or for some r, m and k, the group G ≤ Sym(m) wr Sym(r), preserving a product structure of r direct copies of the natural action of Sym(m) or Alt(m) on k-sets. In this paper we reduce this conjecture to the case that G is an almost simple group with socle a classical group.