Group Extensions, Representations, and the Schur Multiplicator

Abstract

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Isoclinism of crossed modules

“…✷ In Hall (1940), it is proved that every group is isoclinic to a stem group. This property depents on the construction of Schur multiplicator and stem extensions of a group as given in Beyl and Tappe (1982). The same definitions and constructions were given for crossed modules in Vieites and Casas (2002).…”

“…It can be proved by using the related constructions and definitions given in Beyl and Tappe (1982); Vieites and Casas (2002)…”

“…After then, some new results were given in many papers, such as Jones and Wiegold (1974); Modabbernia (2012); Mohammadzadeh et al (2013); Parvaneh et al (2011); Salemkar et al (2008); Tappe (1976). Also the relation between groups and their stem extensions with Schur multiplicators were given in Beyl and Tappe (1982) where they consider the isoclinism of central group extensions. In this work we consider the 2-dimensional group version, called "crossed module", of isoclinism and determine some basic results.…”

On the capability of groups

“…For instance, if G/Z is an aspherical group [6] (such as the free abelian group on two generators, or any knot group, or any other one-relator group whose relator is not a proper power) then im{x) = 0 since in this case H^G/Z) = 0. Alternatively, suppose that Z is a direct summand of G; since in this case the surjection G -» G/Z is split, the induced homomorphism H 3 …”

“…Following [10] we say that a group G is capable if there exists a group H such that G ^ H/Z(H). An account of the basic theory on capability is given in [3]. The capability of finitely generated abelian groups was studied by Baer [1]: such groups are capable if and only if their two highest torsion coefficients agree.…”