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Certain combinatorial topics in group theory

C K Gupta
2013 International Journal of Group Theory  
This article is intended to be a survey on some combinatorial topics in group theory.  ...  The bibliography at the end is neither claimed to be exhaustive, nor is it necessarily connected with a reference in the text.  ...  We consider in F some polynilpotent series of subgroups Let G = x 1 , . . . , x n | r be a group with a single defining relation. Denote by G ij a canonical image of F ij in G.  ... 
doaj:c4572ce1742a42aa9ef168c7a7953e4e fatcat:uxtlgpbgevcp5onzj2kctiqvjy

Page 1242 of Mathematical Reviews Vol. , Issue 87c [page]

1987 Mathematical Reviews  
The group ring of a free polynilpo- tent group is embedded in a Magnus algebra with homogeneous relations on the variables, but this embedding does not correspond to the Magnus embedding for the free group  ...  For a quasinilpotent group, the minimum number of abelian factors in any finite quasicentral series of G is called the quasinilpo- tency class of G.  ... 

Residually finite groups

Wilhelm Magnus
1969 Bulletin of the American Mathematical Society  
They are also [32] , in the case of a rank ^2, residually T nt where T n is the single-relator group (a,b;a») and possibly residually S for a large number of other single-relator groups 5.  ...  ,i n , fe, ii, * • • ,i n positive integers) as the i n th group of the lower central series of Then the quotient group of F with respect to the group (3.1) is called free polynilpotent of class row, (  ... 
doi:10.1090/s0002-9904-1969-12149-x fatcat:gcnbs4cwuncf7gwdtygvsdvpt4

Groups with the same lower central sequence as a relatively free group. I. The groups

Gilbert Baumslag
1967 Transactions of the American Mathematical Society  
The defining relations then give us E*/E$ as an abelian group on the generators 2 X , ûi.o, • ■ -, am,0t ax,l, • • -, am¡x, Z subject to the single defining relation <o = «fa- ).  ...  Let G be a group, let N be a normal subgroup of G and let C be a complement for N in G, i.e., the variety S3 {i.e., V{G) is the intersection of those normal subgroups of G with factor group in S3).  ... 
doi:10.1090/s0002-9947-1967-0217157-3 fatcat:bm36mkb2encyjcbnoxov4futxq

Groups with the same lower Central Sequence as a Relatively Free Group. I The Groups

Gilbert Baumslag
1967 Transactions of the American Mathematical Society  
The defining relations then give us E*/E$ as an abelian group on the generators 2 X , ûi.o, • ■ -, am,0t ax,l, • • -, am¡x, Z subject to the single defining relation <o = «fa- ).  ...  Let G be a group, let N be a normal subgroup of G and let C be a complement for N in G, i.e., the variety S3 {i.e., V{G) is the intersection of those normal subgroups of G with factor group in S3).  ... 
doi:10.2307/1994377 fatcat:wpyqbbqndfhqbhjwbai77vjtiy

Groups with the Same lower Central Sequence as a Relatively Free Group. II. Properties

Gilbert Baumslag
1969 Transactions of the American Mathematical Society  
Whitehead [33] whereby one can effectively determine whether a group with a given single defining relation is free. • • •» a-i> 2.5. We are left with the proof that H is parafree.  ...  We now add to the relations of F the relations x = 1 if x e X, x $ {x1;..., xk} together with the minimal number of relations which will ensure residual nilpotence of the appropriate factor group of F.  ...  ., xn) is free in 2Í33 on xi5..., xn. Now, by Theorem 6.1, P is torsion-free. So V(P) is a free abelian group of finite rank.  ... 
doi:10.2307/1995369 fatcat:6s6owpyfpjha5geu4mzftnrpae

Groups with the same lower central sequence as a relatively free group. II. Properties

Gilbert Baumslag
1969 Transactions of the American Mathematical Society  
Whitehead [33] whereby one can effectively determine whether a group with a given single defining relation is free. • • •» a-i> 2.5. We are left with the proof that H is parafree.  ...  We now add to the relations of F the relations x = 1 if x e X, x $ {x1;..., xk} together with the minimal number of relations which will ensure residual nilpotence of the appropriate factor group of F.  ...  ., xn) is free in 2Í33 on xi5..., xn. Now, by Theorem 6.1, P is torsion-free. So V(P) is a free abelian group of finite rank.  ... 
doi:10.1090/s0002-9947-1969-0245653-3 fatcat:zihqc5eb4zc2pparkfmjacvppi

On transitivity and (non)amenability of Aut(F_n) actions on group presentations [article]

Aglaia Myropolska, Tatiana Nagnibeda
2014 arXiv   pre-print
The question of (non)amenability of Nielsen graphs is of particular interest in relation with the open question about Property (T) for AutF_n, n≥ 4.  ...  For a finitely generated group G the Nielsen graph N_n(G), n≥rank(G), describes the action of the group AutF_n of automorphisms of the free group F_n on generating n-tuples of G by elementary Nielsen moves  ...  Recall that a group G is called polynilpotent ( [Smir] ) if it admits a finite normal series G ≥ G m 1 ≥ G m 1 ,m 2 ≥ ... ≥ 1 where G m 1 is the m 1th member of its lower central series, G m 1 ,m 2 is  ... 
arXiv:1309.0271v3 fatcat:quk6rork7fdzbfwxvx7tdovtje

Varieties of groups and isologisms

N. S. Hekster
1989 Journal of the Australian Mathematical Society  
This is so to speak isoclinism with respect to a certain variety of groups. The equivalence relation isologism partitions the class of all groups into families.  ...  In order to classify solvable groups Philip Hall introduced in 1939 the concept of isoclinism. Subsequently he defined a more general notion called isologism.  ...  This equivalence relation depends on [12] Varieties of groups and isologisms 33 some fixed variety 53 and has the property t h a t the groups in the variety 53 form a single equivalence class.  ... 
doi:10.1017/s1446788700030366 fatcat:ct5buxiezbgrpdn3sofb4rdsgy

Branch rings, thinned rings, tree enveloping rings

Laurent Bartholdi
2006 Israel Journal of Mathematics  
The main examples come from groups acting on trees.  ...  In particular, for every field k we construct a k-algebra K which (1) is finitely generated and infinite-dimensional, but has only finite-dimensional quotients; (2) has a subalgebra of finite codimension  ...  On the one hand, J has trivial image in P, since in ψ n (v ∗ g − 1) and ψ n (w ∗ h − 1) are diagonal matrices with a single non-zero entry, in different coordinates v, w.  ... 
doi:10.1007/bf02773601 fatcat:dqx4ekrzkvdr3jwhpmeormnmjy

General index, 1965-1974: Research announcements

1975 Bulletin of the American Mathematical Society  
Rider, Daniel, A relation between a theorem of Bohr and Sid on sets, 72, 558 , Central idempotents in group algebra, 72, 1000 Rieffel, Marc A., Induced representations of C*-algebras, 78, 605 Rieger, G  ...  equation and hitting probabilities of single points for processes with stationary independent increments, 75, 573 Keynes, Hafvey B., Topological dynamics in coset transformation groups, 72, 1033 Khabbaz  ...  splitting algebras, 73, 106 Taibleson, Mitchell H" Fourier series on the ring of integers in a p-series field, 73, 623 Tait, W.  ... 
doi:10.1090/s0002-9904-1975-13804-3 fatcat:atkshf2s7jfe3o7bgu62xbbbpy