Five dimensional Bieberbach groups with trivial centre

1990 ◽  
Vol 68 (1) ◽  
pp. 191-208 ◽  
Author(s):  
Andrzej Szczepański
Keyword(s):  
Trivial Centre

scholarly journals Commutation semigroups of finite metacyclic groups with trivial centre

2020 ◽  
Vol 100 (3) ◽  
pp. 765-789
Author(s):  
Darien DeWolf ◽  
Charles C. Edmunds
Keyword(s):  
Metacyclic Groups ◽  
Trivial Centre

Torsion in semicomplete nilpotent groups

1983 ◽  
Vol 94 (2) ◽  
pp. 191-202 ◽  
Author(s):  
Thomas A. Fournelle
Keyword(s):  
Nilpotent Groups ◽  
Trivial Centre ◽  
Object Of Study ◽  
Inner Automorphisms

Let Aut G and Inn G denote the group of all automorphisms of the group G and the subgroup of all inner automorphisms of G, respectively. A group G is said to be complete if it has trivial centre and Aut G = Inn G. Examples of such groups abound and they have been the object of study for many years. Following Heineken (8) we call a group G semicomplete if Aut G = Inn G.

A note on the normalizer condition

1973 ◽  
Vol 74 (1) ◽  
pp. 11-15 ◽  
Author(s):  
B. Hartley
Keyword(s):  
Wreath Product ◽  
Normalizer Condition ◽  
Trivial Group ◽  
Trivial Centre

The first example of a non-trivial group satisfying the normalizer condition but having trivial centre was obtained by Heineken and Mohamed (1). In fact, the group they constructed satisfies the stronger condition that all its proper subgroups are subnormal and nilpotent. In this note we use a construction suggested by their approach to obtain such a group G as a subgroup of the restricted wreath product Cp ≀ Cp∞. The fact that G is given as a subgroup of a well known group makes it rather easier to investigate its properties, and in particular to see that its centre Z(G) is trivial.

scholarly journals A class of one-relator groups with centre

1991 ◽  
Vol 44 (2) ◽  
pp. 245-252 ◽  
Author(s):  
James McCool
Keyword(s):  
Two Stage ◽  
One Relator Groups ◽  
Trivial Centre ◽  
Repeated Applications ◽  
Suitable Group

Let the group H have presentation where m ≥ 3, pi ≥ 2 and (pi, pj) = 1 if i ≠ j. We show that H is a one-relator group precisely if H can be obtained from a suitable group 〈a, b; ap = bp〉 by repeated applications of a (two-stage) procedure consisting of applying central Nielsen transformations followed by adjoining a root of a generator. We conjecture that any one-relator group G with non-trivial centre and G/G′ not free abelian of rank two can be obtained in the same way from a suitable group 〈a, b; ap = bp〉.

scholarly journals A characterisation of cyclic subnormal separated A-groups of nilpotent length three

1997 ◽  
Vol 56 (2) ◽  
pp. 243-251
Author(s):  
Muhammad Umar Makarfi
Keyword(s):  
Prime Order ◽  
Separation Condition ◽  
Fitting Subgroup ◽  
Nilpotent Length ◽  
Monolithic Group ◽  
Derived Subgroup ◽  
Trivial Centre ◽  
Image Position

The paper gives a detailed description of all those finite A-groups of nilpotent length three that satisfy the cyclic subnormal separation condition. It is shown that every monolithic group under discussion is an extension of its Fitting subgroup P, which is a homocyclic p-group, by a p′ metabelian subgroup H, where p is a prime. The centraliser of P in H is trivial while the monolith W is equal to ω1(P) and the action of H on W is faithful and irreducible. H is further shown to have non trivial centre and is an extension of its derived subgroup M by a subgroup L such thatfor all primes q where Mq and Lq are the respective Sylow q-subgroups of M and L. The Fitting subgroup of F of H is shown to be M × Z(H), while Z(H) = F ∩ L and every element of L of prime order is in Z(H). Finally it is shown that if ql(q) is the exponent of Mq then every element of order dividing ql(q) in L belongs to Z(H).

scholarly journals A group with trivial centre satisfying the normalizer condition

1968 ◽  
Vol 10 (3) ◽  
pp. 368-376 ◽  
Author(s):  
H Heineken ◽  
I.J Mohamed
Keyword(s):  
Normalizer Condition ◽  
Trivial Centre

scholarly journals Subgroups with centre in HNN groups

1977 ◽  
Vol 24 (3) ◽  
pp. 350-361 ◽  
Author(s):  
A. Karrass ◽  
D. Solitar
Keyword(s):  
Graph Product ◽  
Trivial Centre

AbstractA subgroup with non-trivial centre in a one-relator group is shown to be a treed HNN group (graph product) with infinite cyclic vertices. Moreover, subgroups with non-trivial centre in HNN groups are also examined.

scholarly journals On the centre and residual finiteness of the automorphism group of a group ring

1987 ◽  
Vol 30 (2) ◽  
pp. 207-213 ◽  
Author(s):  
D. A. R. Wallace
Keyword(s):  
Automorphism Group ◽  
Group Ring ◽  
Infinite Order ◽  
The Other ◽  
Group Rings ◽  
Residual Finiteness ◽  
Original Proof ◽  
Residually Finite ◽  
Cyclic Groups ◽  
Trivial Centre

Let G be a group and let Aut(G) be its automorphism group. It is notorious that the properties of Aut (G) do not relate well to the properties of G, perhaps the only twogeneral results being that if G has a trivial centre then the same is true of Aut (G) [2, p.89] and Baumslag's theorem that if G is finitely generated and residually finite then Aut (G) is also residually finite [1, Theorem 1, p. 117]. In the paper we shall attempt tofind analogues of these results for therelationship between the properties of R(G), the group ring of G over a ring R, and the properties of Aut R(G), the automorphism of R(G). We prove that if R(G) has a trivial centre then Aut R(G) has a trivial centre. We establish the analogue, Theorem 2.3, of Baumslag's theorem by ring-theoretic methods; our original proof used properties of group rings, the present simplified proof we owe to the referee. As an example we calculate Aut ℤ(G) in the case that G is the direct product of two cyclic groups, one of infinite order and the other of order 5. This calculation will, it is hoped, give some indication of the difficulties in determining automorphisms of the group ring of an infinite group.

McLain groups over arbitrary rings and orderings

1995 ◽  
Vol 117 (3) ◽  
pp. 439-467 ◽  
Author(s):  
Manfred Droste ◽  
Rüdiger Göbel
Keyword(s):  
Automorphism Group ◽  
Linear Algebra ◽  
Linear Ordering ◽  
Triangular Matrices ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Trivial Centre ◽  
Dense Sets ◽  
The Difference ◽  
Torsion Free

In 1954, McLain [M] applied some well-known arguments of linear algebra on triangular matrices to establish the existence of characteristically simple, locally finitep-groups, now known as McLain groups [Rob, pp. 347–349]. His groups, having a trivial centre, illustrated sharply the difference between finite and locally finitep-groups. The construction of McLain groups depends on the dense linear ordering (ℚ,≤) and a fieldFpofpelements. It was immediately clear that the parametersFp, ℚ of the McLain groupG(Fp, ℚ) could be replaced by other linearly ordered, dense setsSand by other fieldsFwithout doing much harm to the construction. IfFhas characteristic 0, thenG(F, S) is still locally nilpotent but torsion-free. Wilson [W] investigatedG(Fp, S) for other orderings and Roseblade[R] deeply studied the automorphism group AutG(Fp, S) in his dissertation at Cambridge in 1963.

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