On groups occurring as center factor groups

On the capability of groups

“…A characteristic subgroup Z*{G) of G was defined in [2] using the Schur multiplier, and shown to have the property that Z*(G) is trivial if and only if G is capable. We shall call Z*{G) the "epicentre" of G, and give an alternative definition of it.…”

On the nonabelian tensor square and capability of groups of order p 2 q

“…Baer [2] characterized the capable groups which Vol. 97 (2011) On capable groups of order p 2 q 301 are direct sums of cyclic groups; the capable extra-special p-groups were characterized by Beyl et al [3] (only the dihedral group of order 8 and the extraspecial groups of order p 3 and exponent p are capable); they also described the metacyclic groups which are capable. Magidin [15,16], characterized the 2-generated capable p-groups of class two (for odd p, independently obtained in part by Bacon and Kappe [1]).…”

The precise center of a crossed module

“…In [4], Beyl, Felgner and Schmid showed that there is a common approach to capable and unicentral groups. They define for a given group G a central subgroup Z Ã ðGÞ which is the smallest subject to being the image in G of the center of a central extension of G. Moreover, Z Ã ðGÞ is also the smallest central subgroup of G whose factor group is capable: also a group G is capable if and only if Z Ã ðGÞ ¼ 1 and G is unicentral if and only if Z Ã ðGÞ ¼ ZðGÞ.…”