ABSTRACT

13.1 In the foregoing chapter we introduced normal subgroups and the quotient groups associated to these together with a canonical surjective homomorphism from the group to the quotient group (see 12.16). This construction has a kind of “universal property” stating that any group homomorphism of G to another group, such that the normal subgroup of N of G is mapped to zero, can be factorized via the quotient group (see Theorem 13.3). Then there is a natural question: does the canonical projection map associated to a normal subgroup N of G restrict well to any subgroup H of G? The answer to this is given in the Second Isomorphism Theorem (Theorem 13.9). This is point of a more general problem that is to describe the relation between the structural properties of the quotient group G/N and the properties of certain objects (subgroups between N and G) in G. Theorem 13.10 and the Third Isomorphism Theorem 13.12 may be viewed as first rebults in such a theory.