Examples are also supplied, which help illuminate many of the key concepts contained in the notes. Requiring only a modest initial group theory background from graduate and post-graduate students, these notes provide a field guide to the class of finitely generated solvable groups from a combinatorial group theory perspective.
Read more Read less. Upper-division undergraduates and above. Feldman, Choice, Vol. No customer reviews. Share your thoughts with other customers. Write a customer review. Discover the best of shopping and entertainment with Amazon Prime. Prime members enjoy FREE Delivery on millions of eligible domestic and international items, in addition to exclusive access to movies, TV shows, and more. Example 2. The natural question one can ask is whether there is an algorithm deciding if any given presentation represents the trivial group.
In fact, this triviality problem is algorithmically undecidable. Chapter 3 Constructions. The description of an internal semi-direct product suggests a more general construc- tion of a new group out of a suitably linked pair of groups. The new group, their. The specifics are as follows.
It can be readily verified that. A is now the multiplicative infinite cyclic group generated by a, the isomorphic image of the unity element 1 of Z. Example 3. In this case, however, the infinitely many commuting relations cannot be dispensed with, as we shall see later in Example 5. The action of t on the generators of A is:. Although G is generated by two elements, its subgroup A is not finitely generated. In the next section, we will see this group again as an example of a so-called wreath product.
Suppose that A and T are subgroups of a group W. Guided by the previous analysis of the action of the top group on the base group in a wreath product, in this section we demonstrate how to construct wreath products for all pairs of groups. Suppose that A and T are a given pair of groups. Actually, we can carry out a related construction in greater generality, for groups acting on sets. However, the computation in 3. In these terms, the conjugation action of T on B in Sect. In the following example, we view our wreath product internally. Adaptation of 3. Written out in detail, the action is:.
The reader may wish to try and write a presentation for this group W using knowledge about its structure. Remark 3. Just as intuition might suggest, E can be described conveniently in terms of generators and relations as follows. B has an injected copy in this new group. If a group like G of Sect. The group described in the first example, a Baumslag-Solitar group, is built with the method described in the previous section.
It belongs to the family of two-generator, one-relator groups known as BS m, n. Many Baumslag-Solitar groups, including the one below, possess a remarkable property, which will be demonstrated here. So E is isomorphic to one of its proper quotient groups. This is an example of a so-called non-Hopfian group. Following are two examples of ascending HNN-extensions. If both H and K coincide with B, then t acts as an automorphism of B. We can add another stable letter to E1 as follows.
If this procedure is iterated n times, we have effectively added n commuting stable letters to our original base group. There is another procedure known as a generalized HNN-extension, in which n stable letters are added simultaneously. In this case, the stable letters form a free subgroup of rank n of the HNN-extension.
Chapter 4 Representations and a Theorem of Krasner and Kaloujnine. Theorem 4. Every group G has a faithful permutation representation. Consider SG the symmetric group on the set underlying the group G. Next, define a map. Let G be a group and A a subgroup of G. The element of Y coming from the coset Ag is denoted by g and it is called the representative of the coset Ag. Suppose g, z are elements of G. Lemma 4. The following hold: 1.
If G is simple i. Let N denote the core of A. Assertions 4 and 5 are immediate. Recall that the construction of the unrestricted wreath product Sect. Let X be a new set, on which the group A acts. Proposition 4. Then the unique semi-direct normal form for their product in W is. Therefore, the left hand side of 4. Assume that the actions of T on Y and A on X are faithful. This remarkable theorem states that any non-simple group can be viewed as a subgroup of an unrestricted wreath product. In order to prove this result, we begin with a more general setup.
A acts on itself by right multiplication. Hence, by Proposition 4. Chapter 5 The Bieri—Strebel Theorems. At last, we are ready to begin our investigation of finitely generated solvable groups: we are now prepared to prove two theorems of Bieri and Strebel. These provide an important step in the understanding of finitely presented solvable groups.
If both H and K coincide with B, then t acts as an automorphism of B which is not too interesting ; we will be more concerned with the situation in which exactly one of the associated subgroups coincides with B. We begin with an easy but potent lemma. We need to find two elements in G that generate a non-abelian free subgroup of G. We claim that u and v freely generate a free subgroup. First, we check that u, v each have infinite order. Hence, gp u, v is a free subgroup of rank 2. Of course, there are many other choices possible for u and v.
Theorem 5. Let N be a normal subgroup of a finitely presented group G. We will show that any finitely presented group that has the infinite cyclic group as its homomorphic image is an HNN-extension of a finitely generated group. We need to concoct a base group, with stable letter t. In addition, G is finitely presented, so by Lemma 2. Now we use Tietze transformation T1 see Theorem 2.
As it stands, this is an infinite presentation. But since there are finitely many r j , only finitely many values of m are actually needed. Using T1 , we can now throw out the old a1 ,. On the face of it, given the generators for H and K, neither one appears to equal B. However, in truth we know very little about the interactions among the a p,m. Either H or K or both could in fact equal B.
There is a partial converse to Theorem 5. Proposition 5. The proof of Theorem 5. Lemma 5. If H is a subgroup of finite index in a finitely generated group G, then H is finitely generated. Let Y be a right transversal of H in G. As in Sect. Let X be a set of generators for G. Recall Sect. The first coordinate is a product of elements from Y and elements from X, and this product equals g. Since Y and X are both finite, we have found a finite set of generators for H.
Let G be a finitely generated solvable group. Suppose G is not finite. Consider the first term [G, G] of the derived series for G. Either at least one of those cyclic groups is infinite cyclic, or [G, G] is of finite index in G. Let N be the normal closure of C1 ,. We apply the same argument to [G, G], which we know is finitely generated by Lemma 5. Iterate this process down the derived series until we obtain the groups H and N that we seek.
The procedure will end before we exhaust all the terms of the derived series; or else, G would be finite, contrary to our assumption. Let G be an infinite, finitely presented solvable group. Then G contains a subgroup of finite index which is an ascending HNN- extension of a finitely generated solvable group. By the proof of Lemma 5. Gn is finitely generated, by Lemma 5. It follows that if G is finitely presented, Gn is finitely presented, and by Theorem 5. It guarantees that G is always virtually an ascending HNN-extension of a finitely generated group. Next, we revisit Example 3. Example 5.
As in the case of the additive group of rationals, A is not finitely generated but every finitely generated subgroup of it is infinite cyclic. Following the construction used in Sect. For ease of notation, henceforth we will treat M as an internal semi-direct product.
Suppose that M is finitely presented. By Proposition 1. By Theorem 5. But all finitely generated subgroups of A are infinite cyclic. In an ascending HNN- extension, one of the associated subgroups must. The other case is similarly impossible. Hence, M is not finitely presented. Remark 5. If we express the group operation in A as multiplication, as we did in Sect. Chapter 6 Finitely Generated Metabelian Groups. In the remaining two chapters, we turn our attention exclusively to finitely generated metabelian groups. First we introduce some terminology and auxiliary results.
Lemma 6. Suppose that G is a group generated by the set X. If G is finitely generated, [G, G] is finitely generated as a normal subgroup. Let G be a group and A an abelian, normal subgroup of G. Recall that an endomorphism of a group is a homomorphism from the group to itself. If M is an abelian group, then the set of all endomorphisms of M is a unital ring, denoted by End M. Unital ring homomorphisms map the multiplicative identities of the rings to each other. Unless explicitly stated otherwise, the ring homomorphisms encountered here will be unital.
Suppose now that R is a ring with unity. We also recall that if Q is any group, then the set of all functions from Q to Z with finite support,. We denote this not necessarily commutative ring by ZQ and refer to it as the integral group ring of Q. Additively, it is a free abelian group on the set underlying the group Q, and it contains a copy of Z as the infinite cyclic group generated by 1 the function that sends the identity of Q to 1 and all other elements of Q to 0.
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Let A be an abelian, normal subgroup of a finitely generated group G, and suppose that A is finitely generated as a normal subgroup in G. According to Lemma 6.
Thus, A can be viewed as a finitely generated ZQ-module. Let N be a normal subgroup of a finitely generated metabelian group. Then N is finitely generated as a normal subgroup. Let G be a finitely generated metabelian group. By Corollary 6. It follows that N is the normal closure of a finite set; indeed,. Every finitely generated metabelian group is finitely presented as a metabelian group. Let G be a finitely generated group in the class of all metabelian groups. In this class, we can write a presentation. Thus, Corollary 6. A group G is said to satisfy Max-n if every properly ascending chain of normal subgroups of G is finite.
Theorems 6. Hall: Theorem 6. Finitely generated metabelian groups satisfy Max-n. Let G be a finitely generated metabelian group, and consider a properly ascending chain of normal subgroups of G. By Lemma 6. Hence, there exists an index j and finitely many elements x1 ,. There are only countably many isomorphism classes of finitely generated metabelian groups. Let G be an n-generator metabelian group. Then there exists a normal subgroup N of F, the free metabelian group of rank n, such that. Therefore, the number of n-generator metabelian groups is bounded by the number of finite subsets of the countable group F, which is countable.
Consider the group G from Example 3. G is an extension of an abelian group by an abelian group; hence, metabelian. As we shall see in Chap. However, Corollary 6. The reader is encouraged to consult [4, 5] for a more detailed treatment of the procedure. Not all finitely generated metabelian groups are finitely presented see Example 7. A natural question arises regarding whether we can turn Theorem 5. This embedding theorem links the theories of finitely generated metabelian groups and subgroups of finitely presented metabelian groups. Remark 7. Theorem 7. So far, the different universe has a known boundary at the class of center-by-metabelian groups.
The existence of continuously many non-isomorphic finitely generated center-by-metabelian groups see Theorem 1. We start with an example of a finitely generated metabelian group which is not finitely presented, but forming an HNN-extension embeds it into a finitely presented metabelian group. This will give us an indication of how the proof of Theorem 7. Example 7. W is an abelian-by-abelian group; hence, metabelian. W is finitely generated but not finitely presented. Suppose that W is finitely presented.
Then finitely many of these relators will suffice Lemma 2. Thus, W has an infinite cyclic quotient. Moreover, W does not contain a free subgroup of rank 2 and so, by Theorem 5. In either case, B would be finitely generated, a contradiction. Hence, W is not finitely presented.
To produce a finitely presented ascending HNN-extension of W , consider the subgroup of W generated by aat and t. Our aim is to show that in addition to E being finitely generated, it is also finitely presented and metabelian. To achieve this, we think of W in the same way as in Example 3. Consider the external semi-direct product.
Thus, in particular, W standard wreath product of an infinite cyclic group by another. Under this isomorphism,. Thus, using Tietze transformations Theorem 2. We now prove four lemmas, which will play an important part in the proof of the embedding theorem. Lemma 7. Every finitely generated metabelian group can be embedded in a finitely generated metabelian group which is a semi-direct product of an abelian group A by a finitely generated abelian group Q.
Furthermore, A is a finitely generated ZQ-module. By Theorem 4.