Local Near-Rings and Triply Factorized Groups

Local nearrings with generalized quaternion multiplicative group

“…In particular, it was shown in [5] that if R is a local nearring with identity 1 and L R is the subgroup of all non-invertible elements of R, then the set 1+L R is a subgroup of the multiplicative group R * of R acting on L R by left multiplication, so that the semidirect product L R (1+L R ) is a group of the form G = AB = AM = BM with a normal subgroup M and subgroups A and B such that M is isomorphic to L R and A and B are isomorphic to 1 + L R . Thus in many cases the study of local nearrings can be reduced to that of groups of this form, so called triply factorized groups.…”

Factorizations of groups and related topics

“…Theorem 12.2(Hubert 2004 [60] ). Let R be a nearring with identity element and U an admissable subgroup of R. Let N + be a normal subgroup of U + with (U + 1)N ⊆ N .…”

Locally compact (2, 2)‐transformation groups

“…[5], [10], [14], [15]). The reason that a classification of the plenty of such near-rings is not possible can be found in the fact that already the determination of sharply 2-transitive abstract groups remains without reach (cf.…”