A group, G, is said to be polycyclic if it has a descending chain of subgroups
GG1 Gr Gr 1 1,
in which Gi{Gi 1 is cyclic for i 1,2, . . . , r. Such a chain of subgroups is called a
polycyclic series.
If Gi{Gi 1 xxiGi 1y for each i, then G xx1, . . . , xry. Thus, every poly- cyclic group is finitely generated. The sequence, px1, . . . , xrq, is called a polycyclic generating sequence for G.
Subgroups and quotients of polycyclic groups are themselves polycyclic, for if H ¤Gand N G, then
H HXG1 HXGr HXGr 1 1
and
G{N G1N{N GrN{N Gr 1N{N 1,
are polycyclic series for H and G{N respectively,
Lemma 2.8 gives a characterisation of polycyclic groups as a subclass of the soluble groups. First, a necessary and sufficient condition for an abelian group to be polycyclic:
Lemma 2.7. An abelian group is polycyclic if and only if it is finitely generated. Proof. One direction is clear, for every polycyclic group is finitely generated, and so, in particular, every abelian polycyclic group is finitely generated.
Conversely, suppose that A xa1, . . . , ary is abelian. Then
A xa1, . . . , ary xa1, . . . , ar1y xa1, a2y xa1y 1
is a series with cyclic factors, showing A to be polycyclic.
Lemma 2.8. The polycyclic groups are exactly the soluble groups for which every subgroup is finitely generated.
Proof. Suppose thatGis polycyclic. Then, it follows immediately from the definition of a polycyclic group thatGhas a subnormal series with abelian factors, and is hence soluble. If H ¤G, then H is polycyclic and hence finitely generated.
Conversely, suppose thatGis a soluble group for which every subgroup is finitely generated. Let
GG1 Gr Gr 1 1,
be a soluble series for G. By hypothesis, the abelian factors, Gi{Gi 1, are finitely
generated, and hence polycyclic by Lemma 2.7. Thus, the soluble series above may be refined to obtain a polycyclic series for G, by inserting isomorphic copies of a polycyclic series for each factor, Gi{Gi 1.
In contrast to Lemma 2.7, not every finitely generated soluble group is polycyclic. A counterexample is constructed here. Take
A 8
¹
i8
xaiy
x be the automorphism of A defined by
axi ai 1 p8 i 8q.
Then x has infinite order and generates a group, xxy Z, of automorphisms of A. Now form the semidirect product (see Definition 2.10),
GA xxy.
Then G is soluble, and Gis generated by two elements, namely a0 and x. ButG is
not polycyclic as the subgroup, A, of G is not finitely generated. The group, G, is the so-called wreath product,
G xa0yoxxy ZoZ.
Every polycyclic group admits a specific type of finite presentation that allows for efficient structural computation within the group. Finite presentations for polycyclic groups are discussed in the subsections below. Segal (1983) provides an excellent treatise on the beautiful theory of polycyclic groups.
2.2.1
Finite Soluble Groups
It is an easy corollary of Lemma 2.8 that, in the finite case, the properties “poly- cyclic” and “soluble” are equivalent. To prove this directly, argue as follows. Observe that a soluble series for a finite group has finite abelian factors. Therefore, the Ba- sis Theorem for finite abelian groups may then be applied to decompose the finite abelian factors into direct sums of cyclic groups, thereby yielding a refinement of the given soluble series with cyclic factors as required. Conversely, polycyclicity implies solubility.
Thus, every finite soluble group has a subnormal series with cyclic factors. Such a series gives rise to various finite presentations reflecting the polycyclic structure of the group. These presentations are useful because the Word Problem in such presentations can be solved in an algorithmic fashion.
Let G be a finite soluble group. A presentation forG of the form,
xa1, . . . , ar |a pj j wj,j for 1¤j ¤r, aai j wi,j for 1¤i j ¤ry, where
(i) pj is the least prime such that a pj
j P xaj 1, . . . , ary for j r, and aprr is the identity, and
(ii) wi,j is a word in the generatorsai 1, . . . , ar,
shall be called a power-conjugate presentation for G. The generators of G corre- sponding to a1, . . . , ar in this presentation are known as a power-conjugate generat- ing sequence for G. The relations of the first type are called power relations, while those of the second type are calledconjugate relations.
LetGi xai, . . . , aryfor eachi¤r, and defineGr 1 to be the trivial group. The
presentation above is said to be consistent if |Gi{Gi 1|pi for each i. In this case, every element of G can be written uniquely in the normal form aα1
1 aαrr, where 0¤αi pi for i1, . . . , r.
It is straightforward to show that every finite soluble group possesses a con- sistent power-conjugate presentation, and conversely, that every power-conjugate presentation defines a finite soluble group. Given a consistent power-conjugate pre- sentation for a group, there exists an algorithm (the collection algorithm), which, when given an arbitrary word over the power-conjugate generating sequence, de- termines the corresponding normal word. In particular, collection can be used to
compute the normal word which is equal to the product of two given normal words, thus implementing the group multiplication.
Power-conjugate presentations are an effective way of representing finite soluble groups, and, over the past two decades, a considerable body of efficient algorithms has been developed for computing with soluble groups defined in terms of power- conjugate presentations. For a survey of the algorithms currently in use for power- conjugate presentations, the reader is referred to (Holt et al., 2005, Chap. 8), and, for a discussion of computation in soluble permutation groups, see (Seress, 2003, Chap. 7).
2.2.2
Infinite Polycyclic Groups
A generalisation of the power-conjugate presentation is used to represent infinite polycyclic groups.
Let G be a polycyclic group. A presentation forG of the form,
xa1, . . . , ar |aimi wi,i for iP I, aai j wi,j for 1¤i j ¤r, aa 1 i j wi,j for 1¤i j ¤r, iRIy, where (i) I t1, . . . , ru,
(ii) mi ¡1 foriP I, and
(iii) wi,j is of the formwi,j a
pi,j,|i| 1q
|i| 1 a
pi,j,rq
r , with 0¤pi, j, kq mk if k PI. shall be called a polycyclic presentation for G. The generators of G corresponding to a1, . . . , ar in this presentation are known as a polycyclic generating sequence for G, and the values,mi (iP I), are called the correspondingpolycyclic exponents. The
relations of the first type are called power relations, while those of the second and third types are called conjugate relations.
LetGi xai, . . . , aryfor eachi¤r, and defineGr 1 to be the trivial group. The
presentation above is said to be consistent if the quotient, Gi{Gi 1, has order mi whenever i PI, and is infinite whenever iRI. In this case, every element of G can be written uniquely in the normal form aα1
1 aαrr, where 0¤αi mi for iPI. It is straightforward to show that every polycyclic group possesses a consistent polycyclic presentation, and conversely, that every polycyclic presentation defines a polycyclic group. Given a consistent polycyclic presentation for a group, there exists a version of the collection algorithm, which, when given an arbitrary word over the polycyclic generating sequence, determines the corresponding normal word. In particular, as in the case of power-conjugate presentations, collection can be used to compute the normal word which is equal to the product of two given normal words, thus implementing the group multiplication.
Computing with infinite polycyclic groups are a comparatively new topic in com- putational group theory and the number of available algorithms is much smaller than in the case of finite polycyclic groups. For an accessible introduction to the algorith- mic theory of polycyclic groups, the reader is referred to (Sims, 1994, Chap. 9). A practical account of computing with polycyclic groups can be found in (Eick, 2001).