A "power" conjugate equation in the symmetric group

Transactions of the American Mathematical Society, 1987

Using combinatorial methods, we will examine squares of conjugacy classes in the symmetric groups S ν {S_\nu } of all permutations of an infinite set of cardinality ℵ ν {\aleph _\nu } . For arbitrary permutations p ∈ S ν p \in {S_\nu } , we will characterize when each element s ∈ S ν s \in {S_\nu } with finite support can be written as a product of two conjugates of p p , and if p p has infinitely many fixed points, we determine when all elements of S ν {S_\nu } are products of two conjugates of p p . Classical group-theoretical theorems are obtained from similar results.

Discrete Mathematics, 1980

It is shown that there are no transitive rank 3 extensions of the projective linear groups H, PSL(nr, q) G HS PI'L(m, q), for any prime power q and integer m a 3. In the course of the proof ahe diophantine equation 5" + 11 = x2, where m, x are positive integers, arose. As such equations can now be solved completely we had the choice of using number theory or geometry to complete the proof.

Inventiones mathematicae, 2008

Part of this paper was written while the authors were participating in the Workshop on Lie Groups, Representations and Discrete Mathematics at the Institute for Advanced Study (Princeton). It is a pleasure to thank the Institute for its generous hospitality and support.

arXiv: Combinatorics, 2019

Let $\sigma$ be a permutation on $n$ letters. We say that a permutation $\tau$ is an even (resp. odd) $k$th root of $\sigma$ if $\tau^k=\sigma$ and $\tau$ is an even (resp. odd) permutation. In this article, we obtain generating functions for the number of even and odd $k$th roots of permutations. Our result implies know generating functions of Moser and Wyman and also some generating functions for sequences in The On-line Encyclopedia of Integer Sequences (OEIS).

European Journal of Combinatorics, 1987

Pacific Journal of Mathematics, 1981

Using combinatorial methods, we will prove the following theorem on the permutation group S o of a countable set: If a permutation pεS 0 contains at least one infinite cycle then any permutation of S o is a product of three permutations each conjugate to p. Similar results for permutations of uncoutable sets are shown and classical group theoretical results are derived from this.

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.

Journal of Combinatorial Theory, Series A, 2001

dedicated to the memory of paul erdo s, who inspired so many with so much

Permutations - Actes du colloque Paris Juillet 1972, 1972

We give the coefficients of the irreducible characters of the symmetric group that contribute to the number of cycles of a permutation, considered as a class function of the symmetric group.

Tokyo Journal of Mathematics, 2017