Groups Whose Commutator Subgroups are of Order Two

Abstract

If the commutator subgroup of a group G is of order two the commutators of G are invariant and hence every operator of odd order appears in the central of G since such an operator could not be transformed into an operator of even order. It therefore results that when G involves operators of odd order it is the direct product of a group of order 2m and of an abelian group of odd order. It is desirable to exclude direct products in what follows since all except one of the factor groups would be abelian. Hence we shall assume hereafter that the order of G is 2m and that G is not a direct product. The central of G includes the squares of all the operators of G and hence the central quotient group of G is abelian and of type ln, where n is even since the subgroup composed of all of the operators of G which are commutative with two of its iion-commutative operators is of index 4 under G. It is possible to construct as follows a G whose central is an arbitrary abelian group. If tl, t2, * * *, tI is a set of restricted independent generators of this abelian group we divide these operators into distinct pairs when 1 is even or we divide 1 1 of them into distinct pairs when I is odd. In the former case we construct two operators whose squares are the operators of such a pair and that one of these two operators generates the commutator subgroup. Each of these two operators may be assumed to transform the other into itself multiplied by the commutator of order 2. The remaining pairs of independent generators may be assumed to be such that none of them generates the commutator subgroup but that each of the operators of a pair is the square of an operator of G and that two such operators are again non-commutative but are commutative with all of the other operators thus constructed. We thus arrive at a G which has the given abelian group for its central and is not a direct product of two groups since its central is generated by the squares of its operators. When I is odd we may proceed similarly with the exception that the operator which does not appear in a pair may be assumed to be the only one of the set of independent generators which separately