We give a simple proof that any injective self-mapping of an infinite set M can be written as a product of an injection and a permutation of M both having infinitely many infinite orbits (and no others). This implies Ore’s influential theorem that each permutation of M is a commutator, a similar result due to Mesyan for the injections of M , and a result on which injections f of M can be writte...

in this paper, we verify the solvability of free product of finite cyclic groups with topological methods. we use cayley graphs and everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. in particular, using these methods, we prove that the commutator subgroup of $mathbb{z}_{m}*mathbb{z}_{n}$ is free of rank $(m-1)(n-1)$ fo...

We clarify the relationship between the linear commutator and the ordinary commutator by showing that in any variety satisfying a nontrivial idempotent Mal’cev condition the linear commutator is definable in terms of the centralizer relation. We derive from this that abelian algebras are quasi–affine in such varieties. We refine this by showing that if A is an abelian algebra and V(A) satisfies...

‎A gyrogroup is a nonassociative group-like structure modelled on the ‎space of relativistically admissible velocities with a binary ‎operation given by Einstein's velocity addition law‎. ‎In this ‎article‎, ‎we present a few of groups sitting inside a gyrogroup G‎, ‎including the commutator subgyrogroup‎, ‎the left nucleus‎, ‎and the ‎radical of G‎. ‎The normal closure of the commutator subgyr...

The question considered is the effect on the canonical commutation rules of mass renormalization in quantum etectrodynamics. This is investigated by considering the quantum Langevin equation for a charged quantum oscillator interacting with the radiation field via dipole interaction. Explicit expressions for the commutator are obtained for finite cutoff frequency as well as the large-cutoff lim...