2-Capability of 2-Generator 2-Groups of Class Two
- PDF / 329,580 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 46 Downloads / 188 Views
2-Capability of 2-Generator 2-Groups of Class Two F. Gharibi Monfared1 · S. Kayvanfar1
· F. Johari1
Received: 2 July 2019 / Revised: 1 February 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract The aim of this paper is to classify all 2-capable 2-generator 2-groups of class two. Obtaining the structure of the 2-nilpotent multipliers of these 2-groups is the other aim. Keywords Capable group · 2-Capable group · 2-Nilpotent multiplier · 2-Group Mathematics Subject Classification 20C25 · 20D15
1 Introduction A group G is called capable if there exists some group H such that G ∼ = H /Z (H ). Capability of groups was first appeared in [3], where Baer succeeds to characterize all capable abelian groups among the direct sums of cyclic groups. The concept of capability for p-groups is used in the classification of p-groups into isoclinism classes by Hall [7]. The notion of varietal capability with respect to any variety of groups was introduced by Moghaddam and Kayvanfar [13]. Moreover, Burns and Ellis [6] generalized the concept of capability of groups to the varietal capability with respect to the variety of nilpotent groups of class at most c for c ≥ 1. Recall that a group G is called c-capable if G ∼ = H /Z c (H ) for some group H , where Z c (H ) is the c-th term of the upper central series of H for c ≥ 1. As a result, every c-capable group is also 1-capable.
Communicated by Peyman Niroomand.
B
S. Kayvanfar [email protected] F. Gharibi Monfared [email protected] F. Johari [email protected]
1
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
123
F. G. Monfared et al.
When c = 1, 1-capable groups are indeed capable groups. Later in [6], Burns and Ellis generalized the same results of Baer for the capability of finitely generated abelian groups to c-capability. It is shown that there exists a finite 2-group that is capable but not 2-capable [6, Theorem 1.4]. Therefore, the concepts of capability and c-capability for groups are different from each other. Let G be the quotient of a free group F by a normal subgroup R. Then, the 2-nilpotent multiplier of G is defined as the abelian group M(2) (G) ∼ = R ∩ γ3 (F)/[R, F, F], where γ3 (F) = [F, F, F]. This is a lesser extent of the Baer invariant of a group G with respect to the variety of nilpotent groups of class at most 2, which has been introduced in [4]. The 1-nilpotent multiplier of G is more known as the Schur multiplier of G, M(G), and it is much more studied, for instance in [9,15]. Information about the 2-nilpotent multiplier of groups may be used as an instrument in the connection to the 2-capability of groups. Niroomand and Parvizi proved that all extra-special pgroups are capable and 2-capable simultaneously by obtaining explicit structure of the 2-nilpotent multipliers of all extra-special p-groups in [16]. Recently, Niroomand et al. [17] showed that “capability” and “c-capability” are equivalent for these groups. A new classification for the 2-gen
Data Loading...
