International
Tables for Crystallography Volume A Space-group symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 |
International Tables for Crystallography (2016). Vol. A, ch. 1.1, pp. 7-9
Section 1.1.6. Homomorphisms, isomorphisms^{a}Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands |
In order to relate two groups, mappings between the groups that are compatible with the group operations are very useful.
Recall that a mapping from a set A to a set B associates to each an element , denoted by and called the image of a (under ).
Definition. For two groups and , a mapping from to is called a group homomorphism or homomorphism for short, if it is compatible with the group operations in and , i.e. if The compatibility with the group operation is captured in the phrase
The image of the product is equal to the product of the images.
Fig. 1.1.6.1 gives a schematic description of the definition of a homomorphism. For to be a homomorphism, the two curved arrows are required to give the same result, i.e. first multiplying two elements in and then mapping the product to must be the same as first mapping the elements to and then multiplying them.
It follows from the definition of a homomorphism that the identity element of must be mapped to the identity element of and that the inverse of an element must be mapped to the inverse of the image of , i.e. that . In general, however, other elements than the identity element may also be mapped to the identity element of .
In the case where only the identity element of lies in the kernel of , one can conclude that implies and is called an injective homomorphism. In this situation no information about the group is lost and the homomorphism can be regarded as an embedding of into .
The image of any homomorphism from to forms not just a subset, but a subgroup of . It is not required that is all of , but if this happens to be the case, is called a surjective homomorphism.
Examples
Definition. A homomorphism from to is called an isomorphism if and , i.e. if is both injective and surjective. An isomorphism is thus a one-to-one mapping between the elements of and which is also a homomorphism.
Groups and between which an isomorphism exist are called isomorphic groups, this is denoted by .
Isomorphic groups may differ in the way they are realized, but they coincide in their structure. In essence, one can regard isomorphic groups as the same group with different names or labels for the group elements. For example, isomorphic groups have the same multiplication table if the elements are relabelled according to the isomorphism identifying the elements of the first group with those of the second. If one wants to stress that a certain property of a group will be the same for all groups which are isomorphic to , one speaks of as an abstract group.
Examples
The kernel of a homomorphism is always a normal subgroup, since for and one has . The information about the elements in the kernel of is lost after applying , because they are all mapped to the identity element of . More precisely, if , then all elements from the coset are mapped to the same element in , since for one has . Conversely, if elements are mapped to the same element, they have to lie in the same coset, since implies , thus and thus , i.e. . The cosets of therefore partition the elements of according to their images under . This observation is summarized in the following result, which is one of the most powerful theorems in group theory.
Homomorphism theorem
Let be a homomorphism from to with kernel . Then the factor group is isomorphic to the image via the isomorphism .
Examples