The
nonabelian tensor square G[otimes ] G of a group G is generated by the symbols
g[otimes ] h, g,h ∈ G, subject to the relations
$$gg\prime\otimesh=(^gg\prime\otimes^gh)(g\otimesh) and
g\otimeshh\prime-(g\otimesh)(^hg\otimes^hh\prime),$$ for all $g,g\prime,h,h\prime
\in G< / f>, where $^gg\prime=gg\primeg^{−1}$. The nonabelian tensor square
is a special case of the nonabelian tensor product which has its origins in
homotopy theory. It was introduced by R. Brown and J.-L. Loday in [4]
and [5], extending ideas of J.H.C. Whitehead in [10]. The topic
of this paper is the classification of 2-generator 2-groups of class two up to
isomorphism and the determination of nonabelian tensor squares for these
groups.
In this paper, is a metacyclic 2-group of positive type of nilpotency of class at least three. Let be the set of all subsets of all commuting elements of of size two in the form of where and commute and each of order two. The probability that an element of a group fixes a set is considered as one of the extensions of the commutativity degree that can be obtained by some group actions on a set. In this paper, we compute the probability that an element of fixes a set in which acts on a set, by conjugation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.