‎motivated by a celebrated theorem of schur‎, ‎we show that if~$gamma$ is a normal subgroup of the full automorphism group $aut(g)$ of a group $g$ such that $inn(g)$ is contained in $gamma$ and $aut(g)/gamma$ has no uncountable abelian subgroups of prime exponent‎, ‎then $[g,gamma]$ is finite‎, ‎provided that the subgroup consisting of all elements of $g$ fixed by $gamma$ has finite index‎. ‎so...

Non{Abelian duality in relation to supersymmetry is examined. When the action of the isometry group on the complex structures is non{trivial, extended supersymmetry is realized non{locally after duality, using path ordered Wilson lines. Prototype examples considered in detail are, hyper{Kahler metrics with SO(3) isometry and supersymmetric WZW models. For the latter, the natural objects in the ...

1 Multiply transitive groups Theorem 1.1. Let Ω be a finite set and G ≤ Sym(Ω) be 2–transitive. Let N E G be a minimal normal subgroup. Then one of the following holds: (a) N is regular and elementary abelian. (b) N is primitive, simple and not abelian. Proof. First we show that N is unique. Suppose that M is another minimal normal subgroup of G, so N ∩M = {e} and therefore [N,M ] = {e}. Since ...

We give a characterization for fundamental groups of graphs of groups amalgamating cyclic edge subgroups to be cyclic subgroup separable if each pair of edge subgroups has a non-trivial intersection. We show that fundamental groups of graphs of abelian groups amalgamating cyclic edge subgroups are cyclic subgroup separable, hence residually finite, if each edge subgroup is isolated in its conta...

Let G be a finite group and assume that of automorphisms A is acting on such have coprime orders. Recall subgroup H said to TI-subgroup if it has trivial intersection with its distinct conjugates in G. We study the solubility other properties when we certain invariant subgroups are TI-subgroups, precisely all A-invariant subgroups, non-nilpotent non-abelian G, respectively, TI-subgroups.

We generalise an extension theorem for terraces for abelian groups to apply to nonabelian groups with a central subgroup isomorphic to the Klein 4-group V . We also give terraces for three of the non-abelian groups of order a multiple of 8 that have a cyclic subgroup of index 2 that may be used in the extension theorem. These results imply the existence of terraces for many groups that were not...

An ðm; u; k; lÞ relative difference set (RDS) in a group G relative to a subgroup U of order u and index m is a k-element subset R of G such that every element of g 2 G=U has exactly l representations g 1⁄4 r1r 1 2 with r1; r2 2 R and no identity element of U has such a representation. The subgroup U is often called the forbidden subgroup. If k 1⁄4 ul; then the RDS is called semiregular and its...

A complete enumeration of relative difference sets (RDS) with parameters (16, 4, 16, 4) in a group of order 64 with a normal subgroup N of order 4 is given. If N = Z4, three of the eleven abelian groups of order 64, and 23 of the 256 nonabelian groups of order 64 contain (16, 4, 16, 4) RDSs. If N = Z2 × Z2, six of the abelian groups and 194 of the non-abelian groups of order 64 contain (16, 4, ...

The problem of finding R-sequencings for abelian groups of even orders has been reduced to that of finding R∗sequencings for abelian groups of odd orders except in the case when the Sylow 2-subgroup is a non-cyclic non-elementary-abelian group of order 8. We partially address this exception, including all instances when the group has order 8t for t congruent to 1, 2, 3 or 4 (mod 7). As much is ...