arXiv subjects
Publications and source records attributed to Arturo Magidin.
An argument used to show that certain varieties of nilpotent groups have instances of nontrivial dominions is considered, and generalized. The same is done with the argument used to show that there are nontrivial dominions in the variety of metabelian groups, to suggest how this general technique may be used.
A full characterization of when a subgroup $H$ of a group $G$ in a varietal product ${\cal NQ}$ is epimorphically embedded in $G$ (in the variety ${\cal NQ}$) is given. From this, a result of S.~McKay is derived, which states that if ${\cal NQ}$ has instances of nonsurjective epimorphisms, then ${\cal N}$ also has instances of nonsurjective epimorphisms. Two partial converses to McKay's result are also given: when~$G$ is a finite nonabelian simple group; and when~$G$ is finite and ${\cal Q}$ is a product of varieties of nilpotent groups, each of which contains the infinite cyclic group.
Let~$S$ be a finite nonabelian simple group, and let $H$ be a subgroup of $S$. In this work, the dominion (in the sense of Isbell) of $H$ in $S$ in rmVar(S)$ is determined, generalizing an example of B.H. Neumann. A necessary and sufficient condition for $H$ to be epimorphically embedded in $S$ is obtained. These results are then extended to a variety generated by a family of finite nonabelian simple groups.
A necessary and sufficient condition for an element of an algebra (in the sense of Universal Algebra) to be in the dominion of a subalgebra is given, in terms of transferable sets. This criterion is then used to formulate a more wieldy sufficient condition. Finally, some connections to a purely combinatorial setting are outlined.
We show that every nilpotent group of class at most two may be embedded in a central extension of abelian groups with bilinear cocycle. The embedding is shown to depend only on the base group. Some refinements are obtained by considering the cohomological situation explicitly.