Let G be a group and G′ its commutator subgroup. Denote by c G the minimal number such that every element ofG′ can be expressed as a product of at most c G commutators. A group G is called a c-group if c G is finite. For any positive integer n, denote by cn the class of groups with commutator length, c G n. Let Fn,t 〈x1, . . . , xn〉 andMn,t 〈x1, . . . , xn〉 be, respectively, the free nilpotent ...

Upper and lower estimates for the maximal commutator length of a noncentral element elementary subgroup general linear group over skew-field are proved on basis an multiplicative this skew-field.

It is shown that the identity component of the group of all homeo-morphisms of a manifold with boundary is perfect, i.e. equal to its commutator subgroup. Diierences between homeomorphism and diieomorphism groups are exhibited.

It is shown that the commutator subgroup of the fundamental group of a smooth irreducible affine curve over an uncountable algebraically closed field k of positive characteristic is a profinite free group of rank equal to the cardinality of k.

We prove that any element in the group generated by Riordan involutions is product of at most four them. also give a description this subgroup as semidirect special commutator and Klein four-group.

In this paper, by calculating the commutator subgroup of the unit group of finite path algebra k∆ and the unit group abelianized, we explicitly characterize the K1 group of finite dimensional path algebra over an arbitrary field.