A note on multivalued groups

Transformation Groups, 1997

This paper introduces the concept of n-valued groups and studies their algebraic and topological properties. We explore a number of examples. An important class consists of those that we call n-coset groups; they arise as orbit spaces of groups G modulo a group of automorphisms with n elements. However, there are many examples that do not arise from this construction. We see that the theory of n-valued groups is distinct from that of groups with a given automorphism group. There are natural concepts of the action of an n-valued group on a space and of a representation in an algebra of operators. We introduce the (purely algebraic) notion of an n-Hopf algebra and show that the ring of functions on an n-valued group and, in the topological case, the cohomology has an n-Hopf algebra structure. The cohomology algebra of the classifying space of a compact Lie group admits the structure of an n-Hopf algebra, where n is the order of the Weyl group; the homology with dual structure is also an n-Hopf algebra. In general the group ring of an n-valued group is not an n-Hopf algebra but it is for an n-coset group constructed from an abelian group. Using the properties of n-Hopf algebras we show that certain spaces do not admit the structure of an n-valued group and that certain commutative n-valued groups do not arise by applying the n-coset construction to any commutative group.

2008

We implement GAP functions about groups with action on itself and investigate some basic properties of small groups with action on itself of order $<32$.

2006

K-Monographs in Mathematics, 2002

There are many pairs of nonisomorphic geometric objects whose products with a third object are isomorphic. Examples of this sort appeared frequently in the research of K. Kawakubo. This paper discusses some examples that have applications to questions about transformation groups of independent interest.

1998

We study and relate certain actions and extensions involving 2-groups.

Georgian Mathematical Journal, 2017

We introduce the notions of center, singularity and nilpotency (class) of a group with action on itself. Also, we describe a new package GwA for GAP4, including functions checking some properties of groups with action on itself with finite underlying group. As applications of the implemented functions, we give examples of groups with action satisfying Condition 1 stated in [

arXiv: Group Theory, 2019

A group of bijections G acting on a set X is said with fixed points (abbreviated as gaf from the french &quot;groupe {\`a} points fixes&quot;) if any element of G has at least one fixed point in X. The G group is said with a common fixed point (abbreviated as gag for &quot;groupe {\`a} point fixe global&quot;) if there is x $\in$ X fixed by all elements of G. The group G is said fixating if any subgroup of G which is a gaf is automatically a gag. The article explores which groups are fixating. The situation depends on the assumptions made on the group of bijections and on the support set X. For example the group of isometries of the Euclidean space R n is fixating for n 3 but not for n 4. The case of isometries of elliptic and hyperbolic spaces is also considered, as well as that of isometries of some discrete sets. As we will see, the situation depends upon the fact whether the so-called median inequality is satisfied in the ambient space or not. Besides, some of our constructions ...

Unlike set, a multiset is a collection of objects in which repetition of elements is significant. In this paper, the study of classical group theory in the context of multisets is presented.