- Polycyclic Group
- GAP3 Manual: 25 Finite Polycyclic Groups
- Polycyclic groups and topology
- Your Answer
- 25.1 More about Ag Groups

Word Problems, Boone, Cannonito, Lyndon eds. MR Gatterdam : The word problem and power problem in 1-relator groups is primitive recursive submitted for publication. Zbl Clapham : Finitely presented groups with word problem of arbitrary degree of insolubility. London Math. Gatterdam : Embeddings of primitive recursive computable groups submitted for publication. Australian Math. Kleene : Introduction to Metamathematics. Van Nostrand, Princeton, New Jersey Can anybody point me in the right direction? Sign up to join this community.

- 40.1 Polycyclic Generating Systems?
- Liederkreis, Op. 39, No. 12, FrÃ¼hlingsnacht (Spring night), piano/violin.
- On the Theory of Locally Polycyclic Groups.
- 25.2 Construction of Ag Groups.

The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Infinite index subgroup of polycyclic group has strictly lower Hirsch length? Ask Question. Asked 7 years, 11 months ago. Active 7 years, 11 months ago.

Viewed times. Andy Andy 38 3 3 bronze badges. If in a finite group every subgroup is subnormal, then the group is nilpotent. Theorem 23 Let Dm denote the dihedral group of order 2m. We conclude this section by providing some basic structural information for any polycyclic group having a polycyclic generating set of length 2. This co- incides with known results on metacyclic groups [5] and [3]. Corollary 24 Let G be a polycyclic group with a polycyclic generating se- quence of length at most 2. Let G be a polycyclic group with a polycyclic generating sequence g1 , g2. We specialize a result of [13] to show that hJi is generated by a finite normal set of words which depend on an arbitrary but fixed generating set of G and a polycyclic generating sequence of G.

Theorem 25 Let G be a polycyclic group. Putting these steps together we get the following algorithm. GAP has methods for constructing finitely presented groups as needed in step 1. The Polycyclic GAP package [8] can be used to effectively compute with finite and infinite polycyclic group such as finding subgroup needed in step 3.

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Hence step 2 is where the real work is done. Two GAP implementations exist to compute the polycyclic quotient of a finitely presented group [12] and [7].

## Polycyclic Group

If we turn our attention to finitely generated nilpotent groups the problem of step 2 in the algorithm above reduces to using a nilpotent quotient algo- rithm since by Theorem?? In the next section we will use both the theoretical results of Section 3 and examples to compute the nonabelian tensor square of the free nilpotent groups of class 3 and rank n. A GAP implementation of Algorithm 1 helped in this task in that we abel to compute examples of small rank from which relations for the general case could be determined. We use the theoretical results of Section 3 to aid our work.

In our context, we are able to extend Lemma 7 i. Lemma 26 Suppose the group G is nilpotent of class 3. Corollary 27 Suppose the group G is nilpotent of class 3. We use this property in the sequel without further reference. The following case is worth recording. Lemma 28 Suppose that G is a nilpotent group of class at most 3. The Jacobi identity is used several times in the proofs that follow.

## GAP3 Manual: 25 Finite Polycyclic Groups

The following result follows immediately. For the sake of convenience we list a particular instance of Lemma Corollary 31 Suppose that G is a nilpotent group of class at most 3. Similarly, inverses can be pulled out directly from weight four commutators and from weight three commutators if the term inverted is a weight two com- mutator. We use these facts without further comment. Lemma 32 Suppose that G is a nilpotent group of class at most 3.

Lemma 33 Suppose that G is a nilpotent group of class at most 3. Corollary 34 Suppose that G is a nilpotent group of class at most 3. Lemma 35 Suppose that G is a nilpotent group of class at most 3. Substitution into Equation 11 yields the result. Lemma 36 Suppose that G is a nilpotent group of class at most 3. TeX- smith?

## Polycyclic groups and topology

Although i - iii are special cases of iv - vi , we use the former to prove the latter. We now have the machinery available to describe the nonabelian tensor square of the free nilpotent group of class 3 and rank n. Let G be the free nilpotent of class 3 group of rank n generated by g1 ,. How to phrase? In what follows we describe and count a minimal set of generators for such a subgroup A.

Hence we retain as generators of A!

### Your Answer

We again consider two possibilities. We therefore retain! In this revision I am not yet tackling the question of proving isomorphism. I wanted to get at least the Theorem 37 Let G be a free nilpotent group of class 3 and rank n. I will likely need to talk with you about this. Question: presum- ably N should be free nilpo- and 26 the rank f n of A is tent of class 2, not just nilpotent of class 2. On the nonabelian tensor square of a nilpotent group of class two.

Glasgow Math. On the nonabelian tensor square of a 2-Engel group. Basel , 69 5 â€” , Beuerle and Luise-Charlotte Kappe.

### 25.1 More about Ag Groups

Infinite metacyclic groups and their non-abelian tensor squares. Edinburgh Math. On computing the non-abelian tensor squares of the free 2-Engel groups.

erejuxubiwyn.tk Brown, D. Johnson, and E.