We prove that a non-abelian superstable CSA-group has an infinite definable simple subgroup all of whose proper definable subgroups are abelian. This imply in particular that the existence of non-abelian CSAgroup of finite Morley rank is equivalent to the existence of a simple bad group all whose definable proper subgroups are abelian. We give a new proof of a result of E. Mustafin and B. Poiza...

Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2|K|-many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22 |K| -many strictly finer pseudocompact topological group refinements. (C) (2007) Every non-metrizable pseudocompact abelian group has a proper dense pseudoco...

Recently, several works by a number of authors have provided characterizations integral undirected Cayley graphs over generalized dihedral groups and dicyclic groups. We generalize unify these results in two different ways. Firstly, we work arbitrary non-abelian finite admitting an abelian subgroup index 2. Secondly, our main result actually characterizes mixed such groups, the spirit very rece...

We conjecture that one of the main obstacles to creating new non-abelian quantum hidden subgroup algorithms is the correct choice of a transversal.

The just non-(virtually abelian) groups with non-trivial Fitting subgroup are classified. Particular attention is given to those which are virtually nilpotent and examples are given of the interesting phenomena that can occur.

Let G be a finite non-cyclic p-group of order at least p3. If G has an abelian maximal subgroup, or if G has an elementary abelian centre with CG(Z(Φ(G))) 6= Φ(G), then |G| divides |Aut(G)|.

We show that ω-categorical rings with NIP are nilpotent-by-finite. We prove that an ω-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an ω-categorical group with at least one strongly regular type is abelian. Moreover, we get that each ω-categorical, characteristically simple p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the e...

Kraemer has shown that every abelian group of order 2 2 d+ 2 with exponent Jess than 2 2 d+ 3 has a difference set. Generalizing this result, we show that any non-abelian group with a central subgroup of size 2d+ 1 together with an exponent-like condition will have a difference set.

(1) Let G be a group and H ⊂ Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p2 (p a prime number) is abelian and that the centre of any non-abelian group of order p3 has p elements. (2) (?) Let H be a normal subgroup of a p-group G, H 6= {e}. Prove that H∩Z(G) 6= {e}. In particular, one obtains that any normal subgroup with p-elements is con...

Abstract We investigate finite non-Abelian simple groups G for which the projective cover of trivial module coincides with permutation on a subgroup and classify all cases unless is Lie type in defining characteristic.