A combinatorial problem in the symmetric group

The symmetric group - representations, combinatorial algorithms, and symmetric functions

A combinatorial problem arising in group theory

We consider a combinatorial problem occurring naturally in a group theoretical setting and provide a constructive solution in a special case. More precisely, in 1999 the author established a logarithmic bound for the derived length of the quotient of a finite solvable group modulo the second Fitting subgroup in terms of the number of irreducible character degrees of the group. Along the way, in...

On a Combinatorial Problem in Group Theory

Let n be a positive integer or infinity (denote ∞). We denote by W ∗(n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist a positive integer k, and a subset X0 ⊆ X, with 2 ≤ |X0| ≤ n + 1 and a function f : {0, 1, 2, . . . , k} −→ X0, with f(0) 6= f(1) and non-zero integers t0, t1, . . . , tk such that [x0 0 , x t1 1 , . . . , x tk k ] = 1, where xi := f(...

Combinatorial Constructions for Transitive Factorizations in the Symmetric Group

We consider the problem of counting transitive factorizations of permutations; that is, we study tuples (σr , . . . , σ1) of permutations on {1, . . . , n} such that (1) the product σr · · · σ1 is equal to a given target permutation π , and (2) the group generated by the factors σi acts transitively on {1, . . . , n}. This problem is widely known as the Hurwitz Enumeration Problem, since an enc...

A characterization of the symmetric group of prime degree

Let $G$ be a finite group and $Gamma(G)$ the prime graph of $G$‎. ‎Recently people have been using prime graphs to study simple groups‎. ‎Naturally we pose a question‎: ‎can we use prime graphs to study almost‎ ‎simple groups or non-simple groups? In this paper some results in‎ ‎this respect are obtained and as follows‎: ‎$Gcong S_p$ if and only‎ ‎if $|G|=|S_p|$ and $Gamma(G)=Gamma(S_p)$‎, ‎whe...