Narrow Results

Let p be a prime and let B be a p-block of the symmetric group S(n) on n points. Let (D, b_D) be a Sylow B-subgroup of S(n). We consider the fusion system and determine a precise formula for its essential rank. In addition, the p-blocks B which admit a p-local subgroup of S(n) controlling B-fusion are characterized.

Let 𝒫_X and 𝒮_X be the partition monoid and symmetric group on an infinite set X. We show that 𝒫_X may be generated by 𝒮_X together with two (but no fewer) additional partitions, and we classify the pairs α, β ∈ 𝒫_X for which 𝒫_X is generated by 𝒮_X ∪ {α, β}. We also show that 𝒫_X may be generated by the set ℰ_X of all idempotent partitions together with two (but no fewer) additional partitions. In fact, 𝒫_X is generated by ℰ_X ∪ {α, β} if and only if it is generated by ℰ_X ∪ 𝒮_X ∪ {α, β}. We also classify the pairs α, β ∈ 𝒫_X for which 𝒫_X is generated by ℰ_X ∪ {α, β}. Among other results, we show that any countable subset of 𝒫_X is contained in a 4-generated subsemigroup of 𝒫_X, and that the length function on 𝒫_X is bounded with respect to any generating set.

Let $G$ be an almost simple group with socle $A_n$, the alternating group of degree $n$. We prove that there is a unit of order $pq$ in the integral group ring of $G$ if and only if there is an element of that order in $G$ provided $p$ and $q$ are primes greater than $\frac{n}{3}$. We combine this with some explicit computations to verify the prime graph question for all almost simple groups with socle $A_n$ if $n\le 17$.

We show that the theorem by Hemmer and Nakano, on uniqueness of Specht filtration multiplicities, can be proved working entirely with representations of symmetric groups, or Hecke algebras. Furthermore, we give a new proof that Schur algebras are quasi-hereditary provided the characteristic of the field is at least 5. Our tools are some more general results on stratifying systems.

We prove there is no ring with unit group isomorphic to S_n for n ≥ 5 and that there is no ring with unit group isomorphic to A_n for n ≥ 5, n ≠ 8. To prove the non-existence of such a ring, we prove the non-existence of a certain ideal in the group algebra 𝔽₂[G], with G an alternating or symmetric group as above. We also give examples of rings with unit groups isomorphic to S₁, S₂, S₃, S₄, A₁, A₂, A₃, A₄, and A₈. Most of our existence results are well-known, and we recall them only briefly; however, we expect the construction of a ring with unit group isomorphic to S₄ to be new, and so we treat it in detail.

A class of subgroups of order $1,2^{2-1}2!$, $2^{3-1}3!,\dots ,$$2^{([\frac{n}{2}]-1)}[\frac{n}{2}]!$ of even permutations is obtained for symmetric group $S_n$ using the set ${\overrightarrow{B}}_n$ of signed Brauer diagrams. Consequently, every group of order $n(n-1)\dots \left(\right.\left[\right.\frac{n}{2}\left]\right.+1\left)\right.$ has a Sylow$2$-subgroup of order $2^{[\frac{n}{2}]}$.

We verify a conjecture proposed by Qian on character codegrees in the case of symmetric and alternating groups.

It is shown that the multiplicative monoids of Brauer's centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the self-adjoint functor is satisfied. As in a previous paper, of which this is a companion, it is shown that such a symmetric self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices.

We derive relations between theta functions evaluated at period matrices of cyclic covers of order 3 ramified above 3k points.

Rosenbloom and Tsfasman introduced a new metric (RT metric) which is a generalization of the Hamming metric. In this paper we propose a Cayley interconnection network based on the RT metric. We also introduce a new family of Cayley graphs on symmetric groups with respect to the RT metric. We also describe various algebraic and topological properties of these networks and propose optimal routing algorithm.