free solvable group
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2012 ◽  
Vol 22 (08) ◽  
pp. 1240005
Author(s):  
SVETLA VASSILEVA

We show that the Magnus embedding, which embeds the free solvable group Sd, r of rank r and degree d into the wreath product ℤr ≀ Sd-1, r, is a quasi-isometry.


2012 ◽  
Vol 22 (04) ◽  
pp. 1250030
Author(s):  
LUCAS SABALKA ◽  
DMYTRO SAVCHUK

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety AmAn, and let A = {a1,…, ar} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {al+1,…, ar} for some l < r, then S is a subset of a basis for the relatively free group on {a1,…, al}.


2006 ◽  
Vol 45 (4) ◽  
pp. 254-260 ◽  
Author(s):  
E. I. Timoshenko

1989 ◽  
Vol 31 (2) ◽  
pp. 193-194
Author(s):  
Howard Smith

There is no example known of a non-nilpotent, torsion-free group which has all of its subgroups subnormal. It was proved in [3] that a torsion-free solvable group with all of its proper subgroups subnormal and nilpotent is itself nilpotent, but that seems to be the only published result in this area which is concerned specifically with torsion-free groups. Possibly the extra hypothesis that the group be hypercentral is sufficient to ensure nilpotency, though this is certainly not the case for groups with torsion, as was shown in [7]. The groups exhibited in that paper were seen to have hypercentral length ω + 1, and we know from [8] that further restricting the hypercentral length can lead to some positive results. Here we shall prove the following theorem.


1984 ◽  
Vol 27 (4) ◽  
pp. 485-489
Author(s):  
A. H. Rhemtulla ◽  
H. Smith

AbstractA group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.


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