GAP (ANUPQ) - Chapter 2: Mathematical Background and Terminology

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2 Mathematical Background and Terminology
2.1 Basic notions
2.1-1 pc Presentations and Consistency
2.1-2 Exponent-\(p\) Central Series and Weighted pc Presentations
2.1-3 \(p\)-Cover, \(p\)-Multiplicator
2.1-4 Descendants, Capable, Terminal, Nucleus
2.1-5 Laws
2.2 The p-quotient Algorithm
2.2-1 Finding the \(p\)-cover
2.2-2 Imposing the Relations of the fp Group
2.2-3 Imposing Laws
2.3 The p-group generation Algorithm, Standard Presentation, Isomorphism Testing

### 2 Mathematical Background and Terminology

In this chapter we will give a brief description of the mathematical notions used in the algorithms implemented in the ANU `pq` program that are made accessible from GAP through this package. For proofs and details we will point to relevant places in the published literature. Also we will try to give some explanation of terminology that may help to use the "low-level" interactive functions described in Section Low-level Interactive ANUPQ functions based on menu items of the pq program. However, users who intend to use these functions are strongly advised to acquire a thorough understanding of the algorithms from the quoted literature. There is little or no checking done in these functions and naive use may result in incorrect results.

#### 2.1 Basic notions

##### 2.1-1 pc Presentations and Consistency

For details, see e.g. [NNN98].

Every finite \(p\)-group \(G\) has a presentation of the form:

\[ \{a_1,\dots,a_n \mid a_i^p = v_{ii}, 1 \le i \le n, [a_k, a_j] = v_{jk}, 1 \le j < k \le n \}. \]

where \(v_{jk}\) is a word in the elements \(a_{k+1},\dots,a_n\) for \(1 \le j \leq k \le n\).

This is called a power-commutator presentation (or pc presentation or pcp) of \(G\), generators from such a presentation will be referred to as pc generators. In terms of such pc generators every element of \(G\) can be written in a "normal form" \(a_1^{e_1}\dots a_n^{e_n}\) with \(0 \le e_i < p\). Moreover any given product of the generators can be brought into such a normal form using the defining relations in the above presentation as rewrite rules. Any such process is called collection. For the discussion of various collection methods see [LGS90] and [VL90b].

Every \(p\)-group of order \(p^n\) has such a pcp on \(n\) generators and conversely every such presentation defines a \(p\)-group. However a \(p\)-group defined by a pcp on \(n\) generators can be of smaller order \(p^m\) with \(m<n\). A pcp on \(n\) generators that does in fact define a \(p\)-group of order \(p^n\) is called consistent in this manual, in line with most of the literature on the algorithms occurring here. A consistent pcp determines a confluent rewriting system (see `IsConfluent` (Reference: IsConfluent) of the GAP Reference Manual) for the group it defines and for this reason often (in particular in the GAP Reference Manual) such a pcp presentation is also called confluent.

Consistency of a pcp is tantamount to the fact that for any given word in the generators any two collections will yield the same normal form.

Consistency of a pcp can be checked by a finite set of consistency conditions, demanding that collection of the left hand side and of the right hand side of certain equations, starting with subproducts indicated by bracketing, will result in the same normal form. There are 3 types of such equations (that will be referred to in the manual):

\[ \begin{array}{rclrl} (a^n)a &=& a(a^n) &&{\rm (Type 1)} \\ (b^n)a &=& b^{(n-1)}(ba), b(a^n) = (ba)a^{(n-1)} &&{\rm (Type 2)} \\ c(ba) &=& (cb)a &&{\rm (Type 3)} \\ \end{array} \]

See [VL84] for a description of a sufficient set of consistency conditions in the context of the \(p\)-quotient algorithm.

##### 2.1-2 Exponent-\(p\) Central Series and Weighted pc Presentations

For details, see [NNN98].

The (descending or lower) (exponent-)\(p\)-central series of an arbitrary group \(G\) is defined by

\[ P_0(G) := G, P_i(G) := [G, P_{i-1}(G)] P_{i-1}(G)^p. \]

For a \(p\)-group \(G\) this series terminates with the trivial group. \(G\) has \(p\)-class \(c\) if \(c\) is the smallest integer such that \(P_c(G)\) is the trivial group. In this manual, as well as in much of the literature about the `pq`- and related algorithms, the \(p\)-class is often referred to simply by class.

Let the \(p\)-group \(G\) have a consistent pcp as above. Then the subgroups

\[ \langle1\rangle < {\langle}a_n\rangle < {\langle}a_n, a_{n-1}\rangle < \dots < {\langle}a_n,\dots,a_i\rangle < \dots < G \]

form a central series of \(G\). If this refines the \(p\)-central series, we can define the weight function \(w\) for the pc generators by \(w(a_i) = k\), if \(a_i\) is contained in \(P_{k-1}(G)\) but not in \(P_k(G)\).

The pair of such a weight function and a pcp allowing it, is called a weighted pcp.

##### 2.1-3 \(p\)-Cover, \(p\)-Multiplicator

For details, see [NNN98].

Let \(d\) be the minimal number of generators of the \(p\)-group \(G\) of \(p\)-class \(c\). Then \(G\) is isomorphic to a factor group \(F/R\) of a free group \(F\) of rank \(d\). We denote \([F, R] R^p\) by \(R^*\). It can be proved (see e.g. [O'B90]) that the isomorphism type of \(G^* := F/R^*\) depends only on \(G\). \(G^*\) is called the \(p\)-covering group or \(p\)-cover of \(G\), and \(R/R^*\) the \(p\)-multiplicator of \(G\). The \(p\)-multiplicator is, of course, an elementary abelian \(p\)-group; its minimal number of generators is called the (\(p\)-)multiplicator rank.

##### 2.1-4 Descendants, Capable, Terminal, Nucleus

For details, see [New77] and [O'B90].

Let again \(G\) be a \(p\)-group of \(p\)-class \(c\) and \(d\) the minimal number of generators of \(G\). A \(p\)-group \(H\) is a descendant of \(G\) if the minimal number of generators of \(H\) is \(d\) and \(H/P_c(H)\) is isomorphic to \(G\). A descendant \(H\) of \(G\) is an immediate descendant if it has \(p\)-class \(c+1\). \(G\) is called capable if it has immediate descendants; otherwise it is terminal.

Let \(G^* = F/R^*\) again be the \(p\)-cover of \(G\). Then the group \(P_c(G^*)\) is called the nucleus of \(G\). Note that \(P_c(G^*)\) is contained in the \(p\)-multiplicator \(R/R^*\).

It is proved (e.g. in [O'B90]) that the immediate descendants of \(G\) are obtained as factor groups of the \(p\)-cover by (proper) supplements of the nucleus in the (elementary abelian) \(p\)-multiplicator. These are also called allowable.

It is further proved there that every automorphism \(\alpha\) of \(F/R\) extends to an automorphism \(\alpha^*\) of the \(p\)-cover \(F/R^*\) and that the restriction of \(\alpha^*\) to the multiplicator \(R/R^*\) is uniquely determined by \(\alpha\). Each extended automorphism \(\alpha^*\) induces a permutation of the allowable subgroups. Thus the extended automorphisms determine a group \(P\) of permutations on the set \(A\) of allowable subgroups (The group \(P\) of permutations will appear in the description of some interactive functions). Choosing a representative \(S\) from each orbit of \(P\) on \(A\), the set of factor groups \(F/S\) contains each (isomorphism type of) immediate descendant of \(G\) exactly once. For each immediate descendant, the procedure of computing the \(p\)-cover, extending the automorphisms and computing the orbits on allowable subgroups can be repeated. Iteration of this procedure can in principle be used to determine all descendants of a \(p\)-group.

##### 2.1-5 Laws

Let \(l(x_1, \dots, x_n)\) be a word in the free generators \(x_1, \dots, x_n\) of a free group of rank \(n\). Then \(l(x_1, \dots, x_n) = 1\) is called a law or identical relation in a group \(G\) if \(l(g_1, \dots, g_n) = 1\) for any choice of elements \(g_1, \dots, g_n\) in \(G\). In particular, \(x^e = 1\) is called an exponent law, \([[x,y],[u,v]] = 1\) the metabelian law, and \([\dots [[x_1,x_2],x_2],\dots, x_2] = 1\) an Engel identity.

#### 2.2 The p-quotient Algorithm

For details, see [HN80], [NO96] and [VL84]. Other descriptions of the algorithm are given in [Sim94].

The `pq` algorithm successively determines the factor groups of the groups of the \(p\)-central series of a finitely presented (fp) group \(G\). If a bound \(b\) for the \(p\)-class is given, the algorithm will determine those factor groups up to at most \(p\)-class \(b\). If the \(p\)-central series terminates with a subgroup \(P_k(G)\) with \(k < b\), the algorithm will stop with that group. If no such bound is given, it will try to find the biggest such factor group.

\(G/P_1(G)\) is the largest elementary abelian \(p\)-factor group of \(G\) and this can be found from the relation matrix of \(G\) using matrix diagonalisation modulo \(p\). So it suffices to explain how \(G/P_{i+1}(G)\) is found from \(G\) and \(G/P_i(G)\) for some \(i \ge 1\).

This is done, in principle, in two steps: first the \(p\)-cover of \(G_i := G/P_i(G)\) is determined (which depends only on \(G_i\), not on \(G\)) and then \(G/P_{i+1}(G)\) as a factor group of this \(p\)-cover.

##### 2.2-1 Finding the \(p\)-cover

A very detailed description of the first step is given in [NNN98], from which we just extract some passages in order to point to some terms occurring in this manual.

Let \(H\) be a \(p\)-group and \(p^{d(b)}\) be the order of \(H/P_b(H)\). So \(d := d(1)\) is the minimal number of generators of \(H\). A weighted pcp of \(H\) will be called labelled if for each generator \(a_k\), \(k > d\) one relation, having this generator as its right hand side, is marked as definition of this generator.

As described in [NNN98], a weighted labelled pcp of a \(p\)-group can be obtained stepping down its \(p\)-central series.

So let us assume that a weighted labelled pcp of \(G_i\) is given. A straightforward way of of writing down a (not necessarily consistent) pcp for its \(p\)-cover is to add generators, one for each relation which is not a definition, and modify the right hand side of each such relation by multiplying it on the right by one of the new generators -- a different generator for each such relation. Further relations are then added to make the new generators central and of order \(p\). This procedure is called adding tails. A more formal description of it is again given in [NNN98].

It is important to realise that the "new" generators will generate an elementary abelian group, that is, in additive notation, a vector space over the field of \(p\) elements. As said, the pcp of the \(p\)-cover obtained in this way need not be consistent. Since the pcp of \(G_i\) was consistent, applying the consistency conditions to the pcp of the \(p\)-cover, in case the presentation obtained for \(p\)-cover is not consistent, will produce a set of equations between the new generators, that, written additively, are linear equations over the field of \(p\) elements and can hence be used to remove redundant generators until a consistent pcp is obtained.

In reality, to follow this straightforward procedure would be forbiddingly inefficient except for very small examples. There are many ways of a priori reducing the number of "new generators" to be introduced, using e.g. the weights attached to the generators, and the main part of [NNN98] is devoted to a detailed discussion with proofs of these possibilities.

##### 2.2-2 Imposing the Relations of the fp Group

In order to obtain \(G/P_{i+1}(G)\) from the pcp of the \(p\)-cover of \(G_i = G/P_i(G)\), the defining relations from the original presentation of \(G\) must be imposed. Since \(G_i\) is a homomorphic image of \(G\), these relations again yield relations between the "new generators" in the presentation of the \(p\)-cover of \(G_i\).

##### 2.2-3 Imposing Laws

While we have so far only considered the computation of the factor groups of a given fp group by the groups of its descending \(p\)-central series, the \(p\)-quotient algorithm allows a very important variant of this idea: laws can be prescribed that should be fulfilled by the \(p\)-factor groups computed by the algorithm. The key observation here is the fact that at each step down the descending \(p\)-central series it suffices to impose these laws only for a finite number of words. Again for efficiency of the method it is crucial to keep the number of such words small, and much of [NO96] and the literature quoted in this paper is devoted to this problem.

In this form, starting with a free group and imposing an exponent law (also referred to as an exponent check) the `pq` program has, in fact, found its most noted application in the determination of (restricted) Burnside groups (as reported in e.g. [HN80], [NO96] and [VL90a]).

Via a GAP program using the "local" interactive functions of the `pq` program made available through this interface also arbitrary laws can be imposed via the option `Identities` (see 6.2).

#### 2.3 The p-group generation Algorithm, Standard Presentation, Isomorphism Testing

For details, see [New77] and [O'B90].

The \(p\)-group generation algorithm determines the immediate descendants of a given \(p\)-group \(G\) up to isomorphism. From what has been explained in Section Basic notions, it is clear that this amounts to the construction of the \(p\)-cover, the extension of the automorphisms of \(G\) to the \(p\)-cover and the determination of representatives of the orbits of the action of these automorphisms on the set of supplements of the nucleus in the \(p\)-multiplicator.

The main practical problem here is the determination of these representatives. [O'B90] describes methods for this and the `pq` program allows choices according to whether space or time limitations must be met.

As well as the descendants of \(G\), the `pq` program determines their automorphism groups from that of \(G\) (see [O'B95]), which is important for an iteration of the process; this has been used by Eamonn O'Brien, e.g. in the classification of the \(2\)-groups that are now also part of the Small Groups library available through GAP.

A variant of the \(p\)-group generation algorithm is also used to define a standard presentation of a given \(p\)-group. This is done by constructing an isomorphic copy of the given group through a chain of descendants and at each step making a choice of a particular representative for the respective orbit of capable groups. In a fairly delicate process, subgroups of the \(p\)-multiplicator are represented by echelonised matrices and a first among the labels for standard matrices is chosen (this is described in detail in [O'B94]).

Finally, the standard presentation provides a way of testing if two given \(p\)-groups are isomorphic: the standard presentations of the groups are computed, for practical purposes compacted and the results compared for being identical, i.e. the groups are isomorphic if and only if their standard presentations are identical.

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