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...

A finite group is called simple when its only normal subgroups are the trivial subgroup and the whole group. For instance, a finite group of prime size is simple, since it in fact has no non-trivial proper subgroups at all (normal or not). A finite abelian group G not of prime size, is not simple: let p be a prime factor of #G, so G contains a subgroup of order p, which is a normal since G is a...

let $a$ be an abelian topological group and $b$ a trivial topological $a$-module. in this paper we define the second bilinear cohomology with a trivial coefficient. we show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...

An infinite group with supersimple theory has a finite series of definable groups whose factors are infinite and either virtually-FC or virtuallysimple modulo a finite FC-centre. We deduce that a group which is typedefinable in a supersimple theory has a finite series of relatively definable groups whose factors are either abelian or simple groups. In this decomposition, the non-abelian simple ...

Let $G$ be a finite group‎. ‎A subset $X$ of $G$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $X$ do not commute‎. ‎In this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.