Arbitrary groups as two-point stabilisers of symmetric groups acting on partitions

Arbitrary groups as two-point stabilisers of symmetric groups acting on partitions

We give a short, direct proof that given any finite group G there exist positive integers k and l and partitions α1 and α2 of {1, . . . , kl} into l subsets of size k such that (Skl)α1,α2 ∼= G. The method used will also show that given any finite group G there exists a regular bipartite graph whose automorphism group is isomorphic to G.

Non-crossing partitions for arbitrary Coxeter groups

Two-generated Groups Acting on Trees

We study 2-generated subgroups of groups that act on simplicial trees. We show that any generating pair {g, h} of such a subgroup is Nielsen-equivalent to a pair {f, s} where either powers of f and s or powers of f and sfs−1 have a common fixed point if the subgroup 〈g, h〉 is freely indecomposable. Analogous results are obtained for generating pairs of fundamental groups of graphs of groups. So...

On Alternating and Symmetric Groups as Galois Groups

Fix an integer n = 3. We show that the alternating group An appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same when n is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in an Sn–extension (i.e. a Galois extension with the symmetric group Sn as Galois group)...

Mod-two Cohomology of Symmetric Groups as a Hopf Ring

On the cohomology of BS• the second product · is cup product, which is zero for classes supported on disjoint components. The first product ⊙ is the relatively new transfer product first studied by Strickland and Turner [21], (see Definition 3.1). It is akin to the “induction product” in the representation theory of symmetric groups, which dates back to Young and has been in standard use [9, 22...