Nielsen's commutator test for two-generator groups

1993 ◽  
Vol 114 (2) ◽  
pp. 295-301 ◽  
Author(s):  
Narain Gupta ◽  
Vladimir Shpilrain
Keyword(s):  
Free Group ◽  
Final Section ◽  
Commutator Subgroup ◽  
Metabelian Group ◽  
Free Metabelian Group ◽  
Sufficiency Condition ◽  
Inner Automorphism ◽  
Tame Automorphism ◽  
Polynilpotent Group ◽  
Inner Automorphisms

Nielsen [14] gave the following commutator test for an endomorphism of the free group F = F2 = 〈x, y; Ø〉 to be an automorphism: an endomorphism ø: F → F is an automorphism if and only if the commutator [ø(x), ø(y)] is conjugate in F to [x, y]±1. He obtained this test as a corollary to his well-known result that every IA-automorphism of F (i.e. one which fixes F modulo its commutator subgroup) is an inner automorphism. Bachmuth et al. [4] have proved that IA-automorphisms of most two-generator groups of the type F/R′ are inner, and it becomes natural to ask if Nielsen's commutator test remains valid for those groups as well. Durnev[7] considered this question for the free metabelian group F/F″ and confirmed the validity of the commutator test in this case. Here we prove that Nielsen's test does not hold for a large class of F/R′ groups (Theorem 3·1) and, as a corollary, deduce that it does not hold for any non-metabelian solvable group of the form F/R″ (Corollary 3·2). In view of our Theorem 3·1, Nielsen's commutator test in these situations seems to have less appeal than his result that the IA-automorphisms of F are precisely the inner automorphisms of F. We explore some applications of this important result with respect to non-tameness of automorphisms of certain two- generator groups F/R (i.e. automorphisms of F/R which are not induced by those of the free group F). For instance, we show that a two-generator free polynilpotent group F/V, , has non-tame automorphisms except when V = γ2(F) or V = γ3(F), or when V is of the form [yn(U), γ(U)], n ≥ 2 (Theorem 4·2). This complements the results of [9] and [16] rather nicely, and is shown to follow from a more general result (Proposition 4·1). We also include an example of an endomorphism θ: x → xu, y→y of F which induces a non-tame automorphism of F/γ6(F) while the partial derivative ∂(u)/∂(x) is ‘balanced’in the sense of Bryant et al. [5] (Example 4·4). This gives an alternative solution of a problem in [5] which has already been resolved by Papistas [15] in the negative. In our final section, we consider groups of the type F/[R′,F] and, in contrast to groups of the type F/R′, we show that the Nielsen's commutator test does hold in most of these groups (Theorem 5·1). We conclude with a sufficiency condition under which Nielsen's commutator test is valid for a given pair of generating elements ofF modulo [R′,F] (Proposition 5·2).

∀-free metabelian groups

1997 ◽  
Vol 62 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Olivier Chapuis
Keyword(s):  
Free Group ◽  
Metabelian Group ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Explicit Description ◽  
Universal Theory ◽  
Metabelian Groups ◽  
Free Metabelian Group ◽  
Finitely Generated Groups

In 1965, during the first All-Union Symposium on Group Theory, Kargapolov presented the following two problems: (a) describe the universal theory of free nilpotent groups of class m; (b) describe the universal theory of free groups (see [18, 1.28 and 1.27]). The first of these problems is still open and it is known [25] that a positive solution of this problem for an m ≤ 2 should imply the decidability of the universal theory of the field of the rationals (this last problem is equivalent to Hilbert's tenth problem for the field of the rationals which is a difficult open problem; see [17] and [20] for discussions on this problem). Regarding the second problem, Makanin proved in 1985 that a free group has a decidable universal theory (see [15] for stronger results), however, the problem of deriving an explicit description of the universal theory of free groups is open. To try to solve this problem Remeslennikov gave different characterization of finitely generated groups with the same universal theory as a noncyclic free group (see [21] and [22] and also [11]). Recently, the author proved in [8] that a free metabelian group has a decidable universal theory, but the proof of [8] does not give an explicit description of the universal theory of free metabelian groups.

Presentations of the Free Metabelian Group of Rank 2

1994 ◽  
Vol 37 (4) ◽  
pp. 468-472 ◽  
Author(s):  
Martin J. Evans
Keyword(s):  
Free Group ◽  
Metabelian Group ◽  
Primitive Element ◽  
Primitive Elements ◽  
Free Metabelian Group ◽  
Rank 2

AbstractLet F3 denote the free group of rank 3 and M2 denote the free metabelian group of rank 2. We say that x * F3 is a primitive element of F3 if it can be included a in some basis of F3. We establish the existence of presentations such that N does not contain any primitive elements of F3.

ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)

2011 ◽  
Vol 21 (04) ◽  
pp. 595-614 ◽  
Author(s):  
S. LIRIANO ◽  
S. MAJEWICZ
Keyword(s):  
Algebraic Group ◽  
Free Group ◽  
Algebraic Variety ◽  
Commutator Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Geometric Invariants ◽  
Torus Knot ◽  
Irreducible Components ◽  
Irreducible Variety

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom (G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), …, N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim (RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, …, xn〉, and letG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, …, xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, …, xn〉, and A = PSL(2, ℂ), then Dim (RA(G)) = Max {3n, Dim (RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, …, xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)).

Some Applications of Artamonov-Quillen-Suslin Theorems to Metabelian Inner Rank and Primitivity

1994 ◽  
Vol 46 (2) ◽  
pp. 298-307 ◽  
Author(s):  
C. K. Gupta ◽  
N. D. Gupta ◽  
G. A. Noskov
Keyword(s):  
Metabelian Group ◽  
Sufficient Conditions ◽  
Surface Group ◽  
Maximal Rank ◽  
Orientable Surface ◽  
Surface Groups ◽  
Modular Element ◽  
Free Metabelian Group ◽  
One Relator Groups ◽  
Necessary And Sufficient

AbstractFor any variety of groups, the relative inner rank of a given groupG is defined to be the maximal rank of the -free homomorphic images of G. In this paper we explore metabelian inner ranks of certain one-relator groups. Using the well-known Quillen-Suslin Theorem, in conjunction with an elegant result of Artamonov, we prove that if r is any "Δ-modular" element of the free metabelian group Mn of rank n > 2 then the metabelian inner rank of the quotient group Mn/(r) is at most [n/2]. As a corollary we deduce that the metabelian inner rank of the (orientable) surface group of genus k is precisely k. This extends the corresponding result of Zieschang about the absolute inner ranks of these surface groups. In continuation of some further applications of the Quillen-Suslin Theorem we give necessary and sufficient conditions for a system g = (g1,..., gk) of k elements of a free metabelian group Mn, k ≤ n, to be a part of a basis of Mn. This extends results of Bachmuth and Timoshenko who considered the cases k = n and k < n — 3 respectively.

The finite inner automorphism groups of division rings

1995 ◽  
Vol 118 (2) ◽  
pp. 207-213 ◽  
Author(s):  
M. Shirvani
Keyword(s):  
Finite Groups ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Commutative Field ◽  
Division Rings ◽  
Inner Automorphism ◽  
Projective Representations ◽  
Inner Automorphisms ◽  
A Finite Group

Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

ON AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE UNIVERSAL ALGEBRA

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1085-1106 ◽  
Author(s):  
G. MASHEVITZKY ◽  
B. I. PLOTKIN
Keyword(s):  
Lie Algebras ◽  
Universal Algebra ◽  
Automorphism Groups ◽  
Endomorphism Semigroup ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Inner Automorphism ◽  
Groups Of Automorphisms ◽  
Inner Automorphisms

Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

Wreath Product of O*-Groups that is not in O*

1971 ◽  
Vol 14 (3) ◽  
pp. 453-454
Author(s):  
S. V. Modak
Keyword(s):  
Wreath Product ◽  
Cyclic Group ◽  
Metabelian Group ◽  
Ordered Group ◽  
Infinite Cyclic Group ◽  
Free Metabelian Group ◽  
Ordered Groups

It is well known that the wreath product of two ordered groups is an ordered group. In [2] Fuchs asks if the same is true for O*-groups. Here we construct an example to show that the wreath product of an infinite cyclic group with a free metabelian group is not an O*-group.

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