International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.1, pp. 7-9

## Section 1.1.6. Homomorphisms, isomorphisms

B. Souvigniera*

aRadboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands
Correspondence e-mail: souvi@math.ru.nl

### 1.1.6. Homomorphisms, isomorphisms

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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.

 Figure 1.1.6.1 | top | pdf |Schematic description of a homomorphism.

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 .

Definition. Let be a group homomorphism from to .

 (i) The set of elements mapped to the identity element of is called the kernel of , denoted by . (ii) The set is called the image of under .

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

• (i) For the symmetry group 4mm of the square a homomorphism to a cyclic group of two elements is given by and , i.e. by mapping the rotations in 4mm to the identity element of and the reflections to the non-trivial element.

Since every element of is the image of some element of 4mm, is a surjective homomorphism, but it is not injective because the kernel consists of all rotations in 4mm and not only of the identity element.

• (ii) The cyclic group of order n is mapped into the (multiplicative) group of the unit circle in the complex plane by mapping to . As displayed in Fig. 1.1.6.2, the image of under this homomorphism are points on the unit circle which form the corners of a regular n-gon.

 Figure 1.1.6.2 | top | pdf |Cyclic group of order 6 embedded in the group of the unit circle.

This is an injective homomorphism because the smallest with is and in , thus by this homomorphism can be regarded as a subgroup of . It is clear that cannot be surjective, because is an infinite group and the image consists of only finitely many elements.

• (iii) For the additive group of integers and a cyclic group , for every integer q a homomorphism is defined by mapping to , which gives for . This is never an injective homomorphism, because is contained in the kernel of . Whether or not is surjective depends on whether is a generator of . This is the case if and only if n and q have no non-trivial common divisors.

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

 (i) The symmetry group 3m of an equilateral triangle is isomorphic to the group of all permutations of . This can be seen as follows: labelling the corners of the triangle by , each element of 3m gives rise to a permutation of the labels and mapping an element to the corresponding permutation is a homomorphism. The only element fixing all three corners of the triangle is the identity element of 3m, thus the homomorphism is injective. On the other hand, the groups 3m and both have 6 elements, hence the homomorphism is also surjective, and thus it is an isomorphism. (ii) For the symmetry group of the square and its normal subgroup generated by the fourfold rotation, the factor group is isomorphic to a cyclic group of order 2. The trivial coset (containing the rotations in 4mm) corresponds to the identity element , the other coset (containing the reflections) corresponds to . (iii) The real numbers form a group with addition as operation and the positive real numbers form a group with multiplication as operation. The exponential mapping is a homomorphism from to because = . It is an injective homomorphisms because only for [which is the identity element in ] and it is a surjective homomorphism because for any there is an with , namely . The exponential mapping therefore provides an isomorphism from to .

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

 (i) The homomorphism from 4mm to sending the rotations in 4mm to and the reflections to has the group of rotations in 4mm as its kernel. The factor group has the cosets = and as its elements and the homomorphism theorem confirms that mapping to and to is an isomorphism from to . (ii) The homomorphism from the additive group of integers to the cyclic group mapping k to has as its kernel. Since is a surjective homomorphism, the homomorphism theorem states that the factor group is isomorphic to the cyclic group . The operation in the factor group is `addition modulo n'.