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Certain combinatorial topics in group theory

2013
*
International Journal of Group Theory
*

This article is intended to be

doaj:c4572ce1742a42aa9ef168c7a7953e4e
fatcat:uxtlgpbgevcp5onzj2kctiqvjy
*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. ...##
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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. ...##
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Residually finite groups

1969
*
Bulletin of the American Mathematical Society
*

They are also [32] ,

doi:10.1090/s0002-9904-1969-12149-x
fatcat:gcnbs4cwuncf7gwdtygvsdvpt4
*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, ( ...##
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Groups with the same lower central sequence as a relatively free group. I. The groups

1967
*
Transactions of the American Mathematical Society
*

The defining

doi:10.1090/s0002-9947-1967-0217157-3
fatcat:bm36mkb2encyjcbnoxov4futxq
*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). ...##
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Groups with the same lower Central Sequence as a Relatively Free Group. I The Groups

1967
*
Transactions of the American Mathematical Society
*

The defining

doi:10.2307/1994377
fatcat:wpyqbbqndfhqbhjwbai77vjtiy
*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). ...##
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Groups with the Same lower Central Sequence as a Relatively Free Group. II. Properties

1969
*
Transactions of the American Mathematical Society
*

Whitehead [33] whereby

doi:10.2307/1995369
fatcat:6s6owpyfpjha5geu4mzftnrpae
*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. ...##
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Groups with the same lower central sequence as a relatively free group. II. Properties

1969
*
Transactions of the American Mathematical Society
*

Whitehead [33] whereby

doi:10.1090/s0002-9947-1969-0245653-3
fatcat:zihqc5eb4zc2pparkfmjacvppi
*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. ...##
###
On transitivity and (non)amenability of Aut(F_n) actions on group presentations
[article]

2014
*
arXiv
*
pre-print

The question

arXiv:1309.0271v3
fatcat:quk6rork7fdzbfwxvx7tdovtje
*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 ...##
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Varieties of groups and isologisms

1989
*
Journal of the Australian Mathematical Society
*

This is so to speak isoclinism

doi:10.1017/s1446788700030366
fatcat:ct5buxiezbgrpdn3sofb4rdsgy
*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. ...##
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Branch rings, thinned rings, tree enveloping rings

2006
*
Israel Journal of Mathematics
*

The main examples come from

doi:10.1007/bf02773601
fatcat:dqx4ekrzkvdr3jwhpmeormnmjy
*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. ...##
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General index, 1965-1974: Research announcements

1975
*
Bulletin of the American Mathematical Society
*

Rider, Daniel,

doi:10.1090/s0002-9904-1975-13804-3
fatcat:atkshf2s7jfe3o7bgu62xbbbpy
*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. ...