A dihedral calculus for groups

Abstract

The classical method of abelianization is an important tool in combinatorial group theory. It is an algorithm for tbe calculation of the commutator factor group and thus gives a survey of the abelian homomorphic images of a group, since any homomorphism of a group into an abelian group factors through the commutator factor group. A non-commutative analogue of abelianization can be developped if one replaces the abelian groups by the (generalized) dihedral groups; in other words, there exists a method which gives a survey of the (generalized) dihedral homomorphic images of a group. The foundations of this new method will be given in Sections 1 and 2. The most conspicuous difference between abelianization and "dihedralization" is the fact that there is no single substitute for the commutator subgroup; it will be shown that the commutator subgroup has to be replaced by a certain canonical set of normal subgroups. Subsequently, instead of having a single algorithm as in the abelian case, several algorithms have to be carried out. The algorithms, which are described in Section 3, are not more complicated than the corresponding one of abelianization. Notation, preliminaries. X- Y denotes the set difference; if Y consists of a single element y, we write X - y. IX[ denotes the cardinal of the set X. If S is a subset of the group G, (S) denotes the subgroup generated by the set S, and S- 1 denotes the set of all inverse elements of S. If G and H are groups, H < G means that H is a subgroup of G. IfH is a normal subgroup of G and x, y are elements of G such that xyN = yxN, we will occasionally say that x and y commute rood N. An elementary abelian 2-group is a group which contains only elements of order at most two. If G is a group, the subgroup generated by all squares is denoted by G 2. Clearly, G 2 iS the smallest of all normal subgroups which have an elementary abelian 2-group as factor group.