Metabelian groups with a single defining relation and the Magnus embedding

1995 ◽  
Vol 57 (4) ◽  
pp. 414-420
Author(s):  
E. I. Timoshenko
Keyword(s):  
Metabelian Groups ◽  
Defining Relation ◽  
Magnus Embedding

A remark on spherical equations in free metabelian groups

2017 ◽  
Vol 0 (0) ◽  
Author(s):  
Evgeny I. Timoshenko
Keyword(s):  
Metabelian Groups ◽  
Diophantine Problem ◽  
Magnus Embedding

AbstractI. Lysenok and A. Ushakov proved that the Diophantine problem for spherical quadric equations in free metabelian groups is solvable. The present paper proves this result by using the Magnus embedding.

Automorphisms of partially commutative metabelian groups

2020 ◽  
Vol 59 (2) ◽  
pp. 239-259
Author(s):  
E. I. Timoshenko
Keyword(s):  
Metabelian Groups

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

2021 ◽  
pp. 1-36
Author(s):  
ARIE LEVIT ◽  
ALEXANDER LUBOTZKY
Keyword(s):  
Ergodic Theorem ◽  
Abelian Groups ◽  
Wreath Products ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Amenable Groups ◽  
Pointwise Ergodic Theorem ◽  
Lamplighter Group ◽  
Invariant Random Subgroups

Abstract We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

scholarly journals Another Law for the 3-Metabelian Groups

1966 ◽  
Vol 6 (4) ◽  
pp. 512-512
Author(s):  
I. D. Macdonald
Keyword(s):  
Mathematical Society ◽  
Metabelian Groups ◽  
Commutator Identity ◽  
Australian Mathematical Society ◽  
Correct Form ◽  
Image Position

Journal of the Australian Mathematical Society 4 (1964), 452–453The second paragraph should be deleted. The alleged commutator identity (3) is false and is certainly not due to Philip Hall. The correct form isas Dr. N. D. Gupta of Canberra has pointed out to me. According to Professor B. H. Neumann, this identity appeared in his (Professor Neumann's) thesis.Nevertheless the theorem is valid and the proof given is correct.

A non-cyclic one-relator group all of whose finite quotients are cyclic

1969 ◽  
Vol 10 (3-4) ◽  
pp. 497-498 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Normal Subgroup ◽  
Residually Finite ◽  
Defining Relation ◽  
Finite Quotient ◽  
Finite Quotients ◽  
Image Position

Let G be a group on two generators a and b subject to the single defining relation a = [a, ab]: . As usual [x, y] = x−1y−1xy and xy = y−1xy if x and y are elements of a group. The object of this note is to show that every finite quotient of G is cyclic. This implies that every normal subgroup of G contains the derived group G′. But by Magnus' theory of groups with a single defining relation G′ ≠ 1 ([1], §4.4). So G is not residually finite. This underlines the fact that groups with a single defining relation need not be residually finite (cf. [2]).

Right and left engel elements in metabelian groups

1981 ◽  
Vol 9 (12) ◽  
pp. 1295-1306 ◽  
Author(s):  
Luise-Charlotte Kappe
Keyword(s):  
Metabelian Groups ◽  
Engel Elements

scholarly journals Homological finiteness conditions for a class of metabelian groups

2017 ◽  
Vol 50 (1) ◽  
pp. 17-25
Author(s):  
Peter H. Kropholler ◽  
Joseph P. Mullaney
Keyword(s):  
Finiteness Conditions ◽  
Metabelian Groups
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