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

The aim of this paper is to present some properties of the Baer-invariant of a pair of groups with respect to a given variety of groups V . We derive some equalities and inequalities of the Baer-invariant of a pair of finite groups, as long as V is considered to be a Schur-Baer variety. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 259 – 266 ¨ ITAK ˙ c TUB doi: Some properties on the Baer-invariant of a pair of groups and VG -marginal series Mohammad Reza RISMANCHIAN1,∗, Mehdi ARASKHAN2 Department of Mathematics, Shahrekord University, Shahrekord, Iran 2 Department of Mathematics, Yazd Branch, Islamic Azad University, Yazd, Iran 1 Received: • Accepted: • Published Online: • Printed: Abstract: The aim of this paper is to present some properties of the Baer-invariant of a pair of groups with respect to a given variety of groups V . We derive some equalities and inequalities of the Baer-invariant of a pair of finite groups, as long as V is considered to be a Schur-Baer variety. Moreover, we present a relative version of the concept of lower marginal series and give some isomorphisms among VG -marginal factor groups. Also, we conclude a generalized version of the Stallings’ theorem. Key words: Baer-invariant, pair of groups, Schur-Baer variety, VG -nilpotent 1. Introduction and preliminaries Let F∞ be the free group freely generated by the countable set X = {x1 , x2, .} , and V be a subset of F∞ . Let V be the variety of groups defined by the set of laws V . We assume that the reader is familiar with the notions of the verbal subgroup, V (G), and the marginal subgroup, V ∗ (G), associated with the variety of groups V and a given group G (see [14] for more information on varieties of groups). Variety V is called a Schur-Baer variety if for any group G in which the marginal factor group G/V ∗ (G) is finite, then the verbal subgroup V (G) is also finite. Schur [17] proved that the variety of abelian groups is a Schur-Baer variety and Baer [2] showed that a variety defined by outer commutator words carries this property. Let G be any group with a normal subgroup N, then we define [NV∗ G] to be the subgroup of G .