Nilpotent Groups Of Class Research Articles

Existentially closed torsion-free nilpotent groups of class three

We denote by and the classes of torsion-free nilpotent groups of nilpotency class at most two and three, respectively. In this paper we show that most of the known results about existentially closed (e.c.) groups in remain true in : Up to isomorphism, there exist only countably many countable e.c. groups and they are distinguished by the ranks of their centers. An e.c. group is finitely (infinitely) generic if and only if the center has dimension one (≥ω). Apart from trivial exceptions, e.c, algebraically closed, and “closed with respect to systems of equations in one unknown” are equivalent.Let K be a class of groups. A group G ϵ K is called existentially closed in K if G contains a solution of any finite system Σ of equations and inequations with constants in G and one or more unknowns, provided that Σ has a solution in some in K. If Σ may contain equations (in at most n unknowns) only, G is called algebraically closed (n-unknown closed [1]). We use the abbreviations e.c, a.c and n-u.c. throughout. For more background on these terms the reader is referred to [3] and [4]. We also assume a basic knowledge of generic structures. The main group theoretic notions needed here are explained in §2; [2] and [11] are the general references for this field.

On existentially closed and generic nilpotent groups

Letn≧2 be an integer. We prove the following results that are known in casen=2: The upper and the lower central series of an existentially closed nilpotent group of classn coincide. A finitely generic nilpotent group of classn is periodic and the center of a finitely generic torsion-free nilpotent group of classn is isomorphic toQ+, whereas infinitely generic nilpotent groups do not enjoy these properties. We determine the structure of the torsion subgroup of existentially closed nilpotent groups of class 2. Finally we give an algebraic proof that there exist 2κ non-isomorphic existentially closed nilpotent groups of classn in cardinalityK ≧N0.

Properties that characterize classes of nilpotent groups

Obviously we have not covered aU possible ascendant and descendant nilpotent properties. Using (1.1), (1.2), or other methods, and Lemma 1.2 it is possible to provide many more such properties. For exampleCq∼(Cpm×Cpm) is a metabelian group having infinitely many non-isomorphic Sylow-p-subgroups [6]. Thus, having one isomorphism class of Sylow-p-subgroups is a descendant nilpotent property.

Varieties of solvable pro-P-groups

By a variety of pro~ -groups we will mean a class of pro~ -groups closed under the operations of taking subgroups, continuous homomorphic images, and topological Cartesian products. It has been shown in [2] that a variety of solvable discrete groups not containing all metablian groups is a subvariety in ~ Dt'Z 0<,<~ for some /z and C , where ~ is the variety of groups of period g,, ~6 the variety of nilpotent groups of nilpotency class ~$ The purpose of the present article is to prove the analog of this result for varieties of solvable pro-p-groups. In Sec. 3 we will prove THEOREM A~ A variety of solvable pro-p-groups not containing all metabelian groups is a subvariety of the variety ~ c ~ p ~ for some oc and C. In Sec. 4, Theorem B is proved generalizing Theorem A to the case of subvarieties of a product of locally finite or abelian varieties. In Sec. 5 we give a description of finitely generated pro-p-groups of finite rank. i. Differentiations of Completed Group Algebras

The diagonal comultiplication on homology

This paper describes the diagonal comultiplication (or cup coproduct) defined on integral homology modules of groups. Analysis of this coproduct should provide a new method of testing for non-isomorphism of groups which have isomorphic integral homology modules; here, the dimension two coproduct is applied to this problem. The first part (Section 2) is couched in terms of groupnets (Brandt groupoids) and shows two things: that there exists a cup product defined on the integral cohomology of any groupnet, extending that for groups, and that there exists a comultiplication defined on the integral homology of any group, natural up to dimension two, which gives the homology modules the structure of a commutative graded co-ring. In the second part (Sections 3 and 4), this diagonal comultiplication R is constructed to dimension two, and the information it carries about the lower central series of a group G is investigated. Modulo torsion in Hr(G; Z), Rz induces an abelian group homomorphism with cokernel GZ/G3, which distinguishes between large classes of groups, in particular the one-relator groups with non-trivial multiplicator, and the finitely-generated nilpotent groups of class two whose relators are all in the commutator subgroup.

Addendum to "Residual Linearity for Certain Nilpotent Groups"

In this note we offer an example of a residually Q-linear, but not residually finite, nilpotent group whose center is Z E Z. This work rounds off some things which have been considered in [1]. Probably the best known result on residual finiteness for linear groups is Mal'cev's Theorem which says that a finitely generated linear group, over a commutative field, is residually finite. This result gives, in particular, that for finitely generated groups, residual finiteness and residual linearity are equivalent. Suppose now that G is a nilpotent linear group. Then it is quite easy to prove that G is finitely generated if its center is so [1, Lemma 1]. Hence we see that a nilpotent linear group with finitely generated center is residually finite. Thus it seems sensible to ask: Is any residually linear nilpotent group with finitely generated center always residually finite? For nilpotent groups of class 2 the answer is affirmative [1, Corollary]. However the answer is negative for groups of class 3. In fact, for each primep there exists a nilpotent group of class 3 with cyclic center, which is residually K-linear but not residually finite, where K is a field containing the p'-roots of unity, for all n > 1 [1, Theorem I(iii)]. It appears to us that K is exceedingly large and so the above question looks more interesting if we consider Q-linearity. In this direction we prove the following: PROPOSITION I. (i) Let G be a residually Q-linear nilpotent group with cyclic center. Then G is residually finite. (ii) There exists a residually Q-linear, but not residually finite, nilpotent group whose center is Z ED Z. Our notation follows that of [1]. However we need the following. Let G be a group. We say that G is a C-group if there exists a finite subset X of G such that Z1(G) = CG(X). If G/Zi(G) is a C-group, for all i > 0, we say that G is a CO-group. For example, any linear group, over a field, is a C0-group. Then we can prove the following: Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 20D15, 20E25; Secondary 20H20.

A note on induced central extensions

A characterization of induced central extensions which gives an explicit relationship between induced central extensions and n-stem extensions is obtained. Using the characterization, necessary and sufficient conditions for a central extension of an abelian group by a nilpotent group of class n to be a Baer sum of an induced central extension and an extension of class n are obtained.

On the nilpotence class of commutative Moufang loops

AbstractThe nilpotence class of the free commutative Moufang loop on n generators (n &gt; 3) is the maximum allowed by the Bruck-Slaby Theorem, namely n − 1. This is proved by setting up a presentation of an extension of the loop's multiplication group as a nilpotent group of class at most 2n − 2, and then using the Macdonald-Wamsley technique of nilpotent group theory to show that this class is exactly 2n − 2.

Splitting theorems in abelian-by-hypercyclic groups

AbstractIf is a saturated formation of finite soluble groups and G is a finite group whose -residual A is abelian then it is well known that G splits over A and the complements are conjugate. Hartley and Tomkinson (1975) considered the special case of this result in which is the class of nilpotent groups and obtained similar results for abelian-by-hypercentral groups with rank restrictions on the abelian normal subgroup. Here we consider the super-soluble case, obtaining corresponding results for abelian-by-hypercyclic groups.

Commutators and abelian groups

AbstractIf G is a group, then K(G) is the set of commutators of elements of G. C is the class of groups such that G′ = K(G) is the minimal cardinality of any generating set of dG. We prove: Theorem A. Let G be a nilpotent group of class two such that G' is finite and d(G′) &lt; 4.Then G &lt; G.Theorm B. Let G be a finite group such that G′ is elementary abelian of order p3. Then G ∈ C.Theorem C. Let G be a finite group with an elementary abelian Sylow p-subgroup S, of order p2, such that S ⊆ K(G). Then S ⊆K(G).

Symmetric words in nilpotent groups of class ≤ 3

Symmetric words in nilpotent groups of class ≤ 3

Automorphism sequences of free nilpotent groups of class two

In this paper we prove that the automorphism group A(N) of a free nilpotent group N of class 2 and finite rank n is complete, except when n is 1 or 3. Equivalently, the centre of A(N) is trivial and every automorphism of A(N) is inner, provided n ≠ 1 or 3. When n = 3, A(N) has an our automorphism of order 2, so A(A(N)) is a split extension of A(N) by . In this case, A(A(N)) is complete. These results provide some evidence supporting a conjecture of Gilbert Baumslag that the sequencebecomes periodic if N is a finitely generated nilpotent group.

Nilpotency class of a free group of the product of certain varieties

In this paper we calculate the class of a free group of finite rank of the product of two varieties, where the left factor is the variety of all Abelian groups of exponent pn and the right is the variety of all nilpotent groups of class at most c and exponent pt (where c < p).

Note on operations generating the group operations in nilpotent groups of class 3

Let K be a class of groups and let ω(K) denote the set of all such words w(x, y) that the group operations 1, x-1, xy in every group G ∈ K can be expressed as a superposition of w and the projections e1(x, y) = x, e2(x, y) = y.

Automorphism Groups of Torsion-Free Nilpotent Groups of Class 2

Automorphism Groups of Torsion-Free Nilpotent Groups of Class 2

Remarks on a class of 2-generator groups of deficiency zero

Let G be a finitely presented group. A finite presentation P of G is said to have defiency m – n if it defines G with m generators and n relations. The deficiency of G is the maximum of the deficiencies of all the finite presentations P of G. If G is finite the deficiency of G is less than or equal to zero. The only finite two generator groups of deficiency zero that are known are certain metacyclic groups given by Wamsley (1970), a class of nilpotent groups given by Macdonald in (1962) and a class of groups given by Wamsley (1972).

On Dye's condition in nilpotent groups of class 2

On Dye's condition in nilpotent groups of class 2

On soluble min-by-max groups

We say a group G has rain (max) if it satisfies the minimal (maximal) condition on subgroups. A group which is an extension of a group with min by a group with max is said to be a rain-by-max group. A well known result of Mal'cev states that a soluble group has max if and only if its abclian subgroups have max. Cernikov proved the analogous result that a soluble group has min if and only if its abelian subgroups have rain. There is no corresponding result for soluble rain-by-max groups. We give an example of a nilpotent group of class 2 which is not rain-by-max, although all its abelian subgroups are. In many ways our example is typical of the worst that can happen. A soluble G group which is not rain-by-max contains a metabelian, hypercentral subgroup X which is not rain-by-max. Furthermore, there is an epimorphic image of X which is structurally identical with the example mentioned above. Thus an investigation of the structure of certain metabelian subgroups will determine completely whether or not a soluble group is min-by-max. We begin therefore, in § 2, by considering the structure of metabelian groups which are not rain-bymax but are such that none of their abelian subgroups have that property. From our investigations we shall then obtain conditions which are necessary and sufficient for a group to be rain-by-max.

A Nonembedding Theorem for Finite Groups

Let $N$ be the class of nilpotent groups with the following properties: (1) The center of $N,{Z_ \bot }(N)$ is of prime order. (2) There exists an abelian characteristic subgroup $A$ of $N$ such that ${Z_1}(N) \subset A \subseteq {Z_2}(N)$ where ${Z_2}(N)$ is the second term in the upper central series of $N$. The main result shown is the following: $N \in \mathfrak {X}$, then $N$ cannot be an invariant subgroup contained in the Frattini subgroup of a finite group.

A nonembedding theorem for finite groups

Let N N be the class of nilpotent groups with the following properties: (1) The center of N , Z ⊥ ( N ) N,{Z_ \bot }(N) is of prime order. (2) There exists an abelian characteristic subgroup A A of N N such that Z 1 ( N ) ⊂ A ⊆ Z 2 ( N ) {Z_1}(N) \subset A \subseteq {Z_2}(N) where Z 2 ( N ) {Z_2}(N) is the second term in the upper central series of N N . The main result shown is the following: N ∈ X N \in \mathfrak {X} , then N N cannot be an invariant subgroup contained in the Frattini subgroup of a finite group.