Some properties of ultra Hall and Schur pairs

The Generalization of the Stalling’s Theorem

“…Schur [9] proved that the variety of abelian groups is a Schur-Baer variety and Baer [10] showed that a variety defined by outer commutator words carries this property. In 2002, Moghaddam et al [11] proved that for a finite group G, M(G) is finite with respect to a Schur-Baer variety . In the following lemma we prove similar result for the M (G, N) and M(G) with using another technique.…”

“…We call such a series the lower  G -marginal series of N in G. One may also define the upper  Gmarginal series as in studies of Moghaddam et al [11].…”

Some properties on the Baer-invariant of a pair of groups and V_G-marginal series

“…We call such a series the lower V G -marginal series of N in G . One may also define the upper V G -marginal series as in [12] . We say that the normal subgroup…”

“…In 2002 , Moghaddam et al [12] proved that for finite group G , VM (G) and hence VM (G, N ) are finite when V is a Schur-Baer variety. Therefore, throughout the rest of this section we always assume that V is a variety of groups which enjoys the Schur-Baer property.…”

The structure of ν-isologic pairs of groups

“…In [6], Moghaddam et al proved that for finite groups G, ν M (G, M ) is finite, with respect to the Schur-Baer variety ν. We use Zorn's lemma and show that every pair of groups is ν-isologic with a quotient irreducible pair of groups.…”