Análisis del mercado

Group

Let E be any extension of an abelian subgroup, M, (regarded as a subgroup of E) by a group,G. So there exists an epimorphismρ: E ÑGwith kernelM. Forg PG, choose ˆg P E with ρpˆgq g and, for m P M, define mg mˆg. Since M is abelian, this definition is independent of the choice of ˆg, and it defines an action of G on M. In general, this action makes M into a _ZG-module, but if M happens to be a

module over a commutative ringK with unity, and the conjugation actions ofg PG define K-automorphisms of M, then M becomes a KG-module. In particular, this is true with K _Fp in the case when M is an elementary abelian p-group for some prime, p.

Definition 2.12. Let G be a group and M a KG-module for some commutative

ring, K. A KG-module extension of M by G is defined to be a group extension, E, of M by G in which the given KG-module, M, is the same as the KG-module defined by conjugation within E.

Given E as above, the set, tˆg | g P Gu, forms a transversal of M in G. For g, h P G, one has ˆgˆh xghτpg, hq, for some function, τ: GG Ñ M, where the associative law in E implies that, for allg, h, k PG,

τpg, hkq τph, kq τpg, hqk τpgh, kq.

A function τ: GGÑM satisfying this identity is called a 2-cocycle, and the additive group of such functions forms aK-module and is denoted by Z2pG, Mq.

Conversely, it is straightforward to check that, for any τ PZ2pG, Mq, the group E tpg, mq |g P G, mPMuwith multiplication defined by

pg1, m1qpg2, m2q pg1g2, τpg1, g2q mg12 m2q

is a KG-module extension of M by G that defines the 2-cocycle τ on choosing ˆ

g pg,0q.

A general transversal of M in E has the form ˆg pg, δpgqq for a function, δ:GÑM, and it can be checked that this transversal defines the 2-cocycle,τ cδ, where cδ is defined by cδpg, hq δpghq δpgqhδphq. A 2-cocycle of the form, cδ, for a function, δ: GÑM, is called a 2-coboundary, and the additive group of such functions is a K-module and it denoted by B2pG, Mq.

TwoKG-module extensions,E1 andE2of aKG-module,M, byGare said to be

equivalent if there is an isomorphism from E1 toE2 that maps the copy ofM inE1

to the copy of M in E2, and induces the identity map on both M and on G. From

the above discussion, it is not difficult to show that the extensions,E1 and E2 with

respective 2-cocycles, τ1 andτ2 are equivalent if and only if τ1τ2 PB2pG, Mqand,

in particular, an extension splits if and only if its corresponding 2-cocycle belongs to B2_{pG, Mq.}

The quotient K-module, H2_{pG, Mq Z}2_{pG, Mq{B}2_{pG, Mq, is called the second}

cohomology group ofG, M and the associated action. It follows from the discussion above that H2p_{G, M}q _{is in one-one correspondence with the equivalence classes of}

Multiplication

Multiplication is the most fundamental operation that one can perform within a group. In order to design a multiplication algorithm which produces consistent results, a normal form for group elements must be defined.

This chapter contains a detailed description of the multiplication algorithm de- veloped for the class of polycyclic-by-finite groups.

Firstly, the proposed normal form for elements of polycyclic-by-finite groups is introduced. After a brief analysis of the technicalities involved in designing a feasi- ble multiplication method, the data structure used to represent polycyclic-by-finite groups is presented, followed by the multiplication algorithm itself. The chapter con- cludes with a survey of useful functions that follow as straightforward applications of the multiplication method.

The definition of the normal form, and the subsequent theory developed, relies on the presupposition that it is computationally feasible to represent the finite quotient of the polycyclic-by-finite group in question faithfully by a group of permutations or matrices. Specifically, a base and strong generating set data structure for the quotient is required. Thus, it shall hereinafter be assumed that, in all cases, such a representation exists and, for the sake of clarity, the finite quotient shall be viewed as a permutation group.

3.1

Representation of Elements

The preliminary aspects of computing with polycyclic-by-finite groups are discussed in this section.