ON INNER AUTOMORPHISMS AND CENTRAL AUTOMORPHISMS OF NILPOTENT GROUP OF CLASS 2

2011 ◽  
Vol 10 (06) ◽  
pp. 1283-1290 ◽  
Author(s):  
ZAHEDEH AZHDARI ◽  
MEHRI AKHAVAN-MALAYERI
Keyword(s):  
Nilpotent Group ◽  
Simple Form ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Free Rank ◽  
Inner Automorphisms ◽  
Central Automorphisms ◽  
Torsion Free Rank ◽  
Torsion Free

Let G be a group and let Aut c(G) be the group of all central automorphisms of G. Let C* = C Aut c(G)(Z(G)) be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper, we prove that if G is a finitely generated nilpotent group of class 2, then C* ≃ Inn (G) if and only if Z(G) is cyclic or Z(G) ≃ Cm × ℤr where [Formula: see text] has exponent dividing m and r is torsion-free rank of Z(G). Also we prove that if G is a finitely generated group which is not torsion-free, then C* = Inn (G) if and only if G is nilpotent group of class 2 and Z(G) is cyclic or Z(G) ≃ Cm × ℤr where [Formula: see text] has exponent dividing m and r is torsion-free rank of Z(G). In both cases, we show G has a particularly simple form.

scholarly journals ON THE CAPABILITY OF FINITELY GENERATED NON-TORSION GROUPS OF NILPOTENCY CLASS 2

2011 ◽  
Vol 53 (2) ◽  
pp. 411-417 ◽  
Author(s):  
LUISE-CHARLOTTE KAPPE ◽  
NOR MUHAINIAH MOHD ALI ◽  
NOR HANIZA SARMIN
Keyword(s):  
Torsion Group ◽  
Nilpotency Class ◽  
Necessary Condition ◽  
Finitely Generated ◽  
Free Rank ◽  
Torsion Groups ◽  
Necessary And Sufficient ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractA group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group.

GENERALIZED ENDOPRIMAL ABELIAN GROUPS

2006 ◽  
Vol 05 (01) ◽  
pp. 1-17 ◽  
Author(s):  
U. ALBRECHT ◽  
S. BREAZ ◽  
W. WICKLESS
Keyword(s):  
Abelian Group ◽  
Simple Form ◽  
Abelian Groups ◽  
Free Rank ◽  
Torsion Groups ◽  
Mixed Groups ◽  
Torsion Free Rank ◽  
Torsion Free ◽  
Nil Radical ◽  
Natural Way

An n-ary endofunction on an abelian group G is a function f : Gn → G such that f(θg1,…,θgn) = θ f(g1,…,gn) for all endomorphisms θ of G. A group G is endoprimal if, for each natural number n, each n-ary endofunction has the following simple form: [Formula: see text] for some collection of integers {li : 1 ≤ i ≤ n}. The notion of endoprimality arises from universal algebra in a natural way and has been applied to the study of abelian groups in papers Davey and Pitkethly (97), Kaarli and Marki (99) and Göbel, Kaarli, Marki, and Wallutis (to appear). These papers make the case that the notion of endoprimality can give rise to interesting and tractable classes of abelian groups. We continue working along these lines, adapting our definition to make it more suitable for working with general classes of abelian groups. We study generalized endoprimal (ge) abelian groups. Here every n-ary endofunction is required to be of the form [Formula: see text] for some collection of central endomorphisms {λi : 1 ≤ i ≤ n} of G. (Note that such a sum is always an endofunction.) We characterize generalized endoprimal abelian groups in a number of cases, in particular for torsion groups, torsion-free finite rank groups G such that E(G) has zero nil radical, and self-small mixed groups of finite torsion-free rank.

scholarly journals Finitely generated groups with virtually free automorphism groups

1995 ◽  
Vol 38 (3) ◽  
pp. 475-484 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Finitely Generated ◽  
Full Automorphism Group ◽  
Finitely Generated Groups ◽  
Special Cases ◽  
Central Quotient ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

It is shown that the full automorphism group of a finitely generated group G is virtually free if and only if the center Z(G) is finitely generated of torsion-free rank r at most two and, depending on the value of r, the central quotient G/Z(G) belongs to one of three precisely defined classes of virtually free groups. Some consequences and special cases are also discussed.

Genus of nilpotent groups of Hirsch length six

1995 ◽  
Vol 117 (3) ◽  
pp. 431-438 ◽  
Author(s):  
Charles Cassidy ◽  
Caroline Lajoie
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Torsion Free

AbstractIn this paper, we characterize the genus of an arbitrary torsion-free finitely generated nilpotent group of class two and of Hirsch length six by means of a finite number of arithmetical invariants. An algorithm which permits the enumeration of all possible genera that can occur under the conditions above is also given.

On the Square Subgroup of a Mixed SI-Group

2018 ◽  
Vol 61 (1) ◽  
pp. 295-304 ◽  
Author(s):  
R. R. Andruszkiewicz ◽  
M. Woronowicz
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Negative Solution ◽  
New Class ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free ◽  
Group Modulo

AbstractThe relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.

scholarly journals Finitely generated nilpotent group C*-algebras have finite nuclear dimension

2018 ◽  
Vol 2018 (738) ◽  
pp. 281-298 ◽  
Author(s):  
Caleb Eckhardt ◽  
Paul McKenney
Keyword(s):  
Irreducible Representation ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Irreducible Representations ◽  
Finitely Generated ◽  
C Algebra ◽  
C Algebras ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
Torsion Free

Abstract We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its ordered K-theory and is approximately subhomogeneous. We observe that all C*-algebras generated by faithful irreducible representations of finitely generated, torsion free nilpotent groups satisfy the universal coefficient theorem.

scholarly journals The torsion-free rank of homology in towers of soluble pro-p groups

2017 ◽  
Vol 219 (2) ◽  
pp. 817-834 ◽  
Author(s):  
Martin R. Bridson ◽  
Dessislava H. Kochloukova
Keyword(s):  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free
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