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 [\varphi] from a set A to a set B associates to each [a \in A] an element [b \in B], denoted by [\varphi(a)] and called the image of a (under [\varphi]).

Definition. For two groups [{\cal G}] and [{\cal H}], a mapping [\varphi] from [{\cal G}] to [{\cal H}] is called a group homomorphism or homomorphism for short, if it is compatible with the group operations in [{\cal G}] and [{\cal H}], i.e. if [\ispecialfonts \varphi({\sfi g} {\sfi g}') = \varphi({\sfi g}) \varphi({\sfi g}') \ {\rm for\ all }\ {\sfi g}, {\sfi g}' \ {\rm in }\ {\cal G}. ]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[link] gives a schematic description of the definition of a homomorphism. For [\varphi] to be a homomorphism, the two curved arrows are required to give the same result, i.e. first multiplying two elements in [{\cal G}] and then mapping the product to [{\cal H}] must be the same as first mapping the elements to [{\cal H}] and then multiplying them.

[Figure 1.1.6.1]

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 [{\cal G}] must be mapped to the identity element of [{\cal H}] and that the inverse [\ispecialfonts{\sfi g}^{-1}] of an element [\ispecialfonts{\sfi g} \in {\cal G}] must be mapped to the inverse of the image of [\ispecialfonts{\sfi g}], i.e. that [\ispecialfonts\varphi({\sfi g}^{-1}) = \varphi({\sfi g})^{-1}]. In general, however, other elements than the identity element may also be mapped to the identity element of [{\cal H}].

Definition. Let [\varphi] be a group homomorphism from [{\cal G}] to [{\cal H}].

  • (i) The set [\ispecialfonts\{ {\sfi g} \in {\cal G} \mid \varphi({\sfi g}) = {\sfi e} \}] of elements mapped to the identity element of [{\cal H}] is called the kernel of [\varphi], denoted by [\ker \varphi].

  • (ii) The set [\ispecialfonts\varphi({\cal G}): = \{ \varphi({\sfi g}) \mid {\sfi g} \in {\cal G} \}] is called the image of [{\cal G}] under [\varphi].

In the case where only the identity element of [{\cal G}] lies in the kernel of [\varphi], one can conclude that [\ispecialfonts\varphi({\sfi g}) = \varphi({\sfi g}')] implies [\ispecialfonts{\sfi g} = {\sfi g}'] and [\varphi] is called an injective homomorphism. In this situation no information about the group [{\cal G}] is lost and the homomorphism [\varphi] can be regarded as an embedding of [{\cal G}] into [{\cal H}].

The image [\varphi({\cal G})] of any homomorphism from [{\cal G}] to [{\cal H}] forms not just a subset, but a subgroup of [{\cal H}]. It is not required that [\varphi({\cal G})] is all of [{\cal H}], but if this happens to be the case, [\varphi] is called a surjective homomorphism.

Examples

  • (i) For the symmetry group 4mm of the square a homomorphism [\varphi] to a cyclic group [\ispecialfonts{\cal C}_2 = \{ {\sfi e}, {\sfi g} \}] of two elements is given by [\ispecialfonts\varphi({1}) = \varphi({4}^+) = \varphi({2}) = \varphi({4}^{-}) = {\sfi e}] and [\varphi({m}_{10}) =] [\ispecialfonts \varphi({m}_{01}) = \varphi({m}_{11}) = \varphi({m}_{1\bar1}) = {\sfi g}], i.e. by mapping the rotations in 4mm to the identity element of [{\cal C}_2] and the reflections to the non-trivial element.

    Since every element of [{\cal C}_2] is the image of some element of 4mm, [\varphi] 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 [\ispecialfonts{\cal C}_n = \{ {\sfi e}, {\sfi g}, {\sfi g}^2, \ldots, {\sfi g}^{n-1} \}] of order n is mapped into the (multiplicative) group [S^1] of the unit circle in the complex plane by mapping [\ispecialfonts{\sfi g}^k] to [\exp(2\pi i k / n)]. As displayed in Fig. 1.1.6.2[link], the image of [{\cal C}_n] under this homomorphism are points on the unit circle which form the corners of a regular n-gon.

    [Figure 1.1.6.2]

    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 [k\,\gt\, 0] with [\exp({2\pi i k / n}) = 1] is [k = n] and [\ispecialfonts{\sfi g}^n = {\sfi e}] in [{\cal C}_n], thus by this homomorphism [{\cal C}_n] can be regarded as a subgroup of [S^1]. It is clear that [\varphi] cannot be surjective, because [S^1] is an infinite group and the image [\varphi({\cal C}_n)] consists of only finitely many elements.

  • (iii) For the additive group [(\bb Z, +)] of integers and a cyclic group [\ispecialfonts{\cal C}_n = \{ {\sfi e}, {\sfi g}, {\sfi g}^2, \ldots, {\sfi g}^{n-1} \}], for every integer q a homomorphism [\varphi] is defined by mapping [1 \in \bb Z] to [\ispecialfonts{\sfi g}^q], which gives [\ispecialfonts\varphi(a) = {\sfi g}^{qa}] for [a \in \bb Z]. This is never an injective homomorphism, because [n{\bb Z} = \{ n a \mid a \in {\bb Z} \}] is contained in the kernel of [\varphi]. Whether or not [\varphi] is surjective depends on whether [\ispecialfonts{\sfi g}^q] is a generator of [{\cal C}_n]. This is the case if and only if n and q have no non-trivial common divisors.

Definition. A homomorphism [\varphi] from [{\cal G}] to [{\cal H}] is called an isomorphism if [\ispecialfonts\ker \varphi = \{ {\sfi e} \}] and [\varphi({\cal G}) = {\cal H}], i.e. if [\varphi] is both injective and surjective. An isomorphism is thus a one-to-one mapping between the elements of [{\cal G}] and [{\cal H}] which is also a homomorphism.

Groups [{\cal G}] and [{\cal H}] between which an isomorphism exist are called isomorphic groups, this is denoted by [{\cal G} \cong {\cal H}].

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 [{\cal G}] will be the same for all groups which are isomorphic to [{\cal G}], one speaks of [{\cal G}] as an abstract group.

Examples

  • (i) The symmetry group 3m of an equilateral triangle is isomorphic to the group [S_3] of all permutations of [\{1, 2, 3\}]. This can be seen as follows: labelling the corners of the triangle by [1, 2, 3], 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 [S_3] both have 6 elements, hence the homomorphism is also surjective, and thus it is an isomorphism.

  • (ii) For the symmetry group [{\cal G} =4mm] of the square and its normal subgroup [{\cal H}] generated by the fourfold rotation, the factor group [{\cal G} / {\cal H}] is isomorphic to a cyclic group [\ispecialfonts{\cal C}_2 = \{ {\sfi e}, {\sfi g} \}] of order 2. The trivial coset (containing the rotations in 4mm) corresponds to the identity element [\ispecialfonts{\sfi e}], the other coset (containing the reflections) corresponds to [\ispecialfonts{\sfi g}].

  • (iii) The real numbers [{\bb R}] form a group with addition as operation and the positive real numbers [{\bb R}_{> 0} =] [ \{ x \in {\bb R} \mid x> 0 \}] form a group with multiplication as operation. The exponential mapping [x \,\mapsto\, \exp(x)] is a homomorphism from [({\bb R}, +)] to [({\bb R}_{> 0}, \cdot)] because [\exp({x+y})] = [ \exp(x) \cdot \exp(y)]. It is an injective homomorphisms because [\exp(x) = 1] only for [x = 0] [which is the identity element in [({\bb R}, +)]] and it is a surjective homomorphism because for any [y\,\gt\, 0] there is an [x \in {\bb R}] with [\exp(x) = y], namely [x = \log(y)]. The exponential mapping therefore provides an isomorphism from [({\bb R}, +)] to [({\bb R}_{> 0}, \cdot)].

The kernel of a homomorphism [\varphi] is always a normal sub­group, since for [\ispecialfonts{\sfi h} \in \ker \varphi] and [\ispecialfonts{\sfi g} \in {\cal G}] one has [\ispecialfonts\varphi({\sfi g} {\sfi h} {\sfi g}^{-1}) =] [\ispecialfonts\varphi({\sfi g}) \varphi({\sfi h}) \varphi({\sfi g}^{-1}) = \varphi({\sfi g}) \varphi({\sfi g}^{-1}) = {\sfi e}]. The information about the elements in the kernel of [\varphi] is lost after applying [\varphi], because they are all mapped to the identity element of [{\cal H}]. More precisely, if [{\cal N} = \ker \varphi], then all elements from the coset [\ispecialfonts{\sfi g} {\cal N}] are mapped to the same element [\ispecialfonts\varphi({\sfi g})] in [{\cal H}], since for [\ispecialfonts{\sfi n} \in {\cal N}] one has [\ispecialfonts\varphi({\sfi g} {\sfi n}) =] [\ispecialfonts \varphi({\sfi g}) \varphi({\sfi n}) = \varphi({\sfi g})]. Conversely, if elements are mapped to the same element, they have to lie in the same coset, since [\ispecialfonts\varphi({\sfi g}) = \varphi({\sfi g}')] implies [\ispecialfonts\varphi({\sfi g}^{-1} {\sfi g}') = {\sfi e}], thus [\ispecialfonts{\sfi g}^{-1} {\sfi g}' \in {\cal N}] and thus [\ispecialfonts{\sfi g}^{-1} {\sfi g}' {\cal N} = {\cal N}], i.e. [\ispecialfonts{\sfi g} {\cal N} = {\sfi g}' {\cal N}]. The cosets of [{\cal N}] therefore partition the elements of [{\cal G}] according to their images under [\varphi]. This observation is summarized in the following result, which is one of the most powerful theorems in group theory.

Homomorphism theorem

Let [\varphi] be a homomorphism from [{\cal G}] to [{\cal H}] with kernel [\ker \varphi = ] [{\cal N}\ \underline\triangleleft\ {\cal G}]. Then the factor group [{\cal G} / {\cal N}] is isomorphic to the image [\varphi({\cal G})] via the isomorphism [\ispecialfonts{\sfi g} {\cal N} \,\mapsto \,\varphi({\sfi g})].

Examples

  • (i) The homomorphism [\varphi] from 4mm to [\ispecialfonts{\cal C}_2 = \{ {\sfi e}, {\sfi g} \}] sending the rotations in 4mm to [\ispecialfonts{\sfi e} \in {\cal C}_2] and the reflections to [\ispecialfonts{\sfi g} \in {\cal C}_2] has the group [\ispecialfonts{\cal N} = \langle {4} \rangle] of rotations in 4mm as its kernel. The factor group [4mm/{\cal N}] has the cosets [{\cal N}] = [ \{ {1}, {4}^+, {2}, {4}^{-} \}] and [{m}_{10} {\cal N} = \{ {m}_{10}, {m}_{01}, {m}_{11}, {m}_{1\bar1} \}] as its elements and the homomorphism theorem confirms that mapping [{\cal N}] to [\ispecialfonts{\sfi e} \in {\cal C}_2] and [{m}_{10} {\cal N}] to [\ispecialfonts{\sfi g} \in {\cal C}_2] is an isomorphism from [4mm/{\cal N}] to [{\cal C}_2].

  • (ii) The homomorphism [\varphi] from the additive group [(\bb Z, +)] of integers to the cyclic group [\ispecialfonts{\cal C}_n = \langle {\sfi g} \rangle] mapping k to [\ispecialfonts{\sfi g}^k] has [{\cal N} = n{\bb Z} = \{ n a \mid a \in {\bb Z} \}] as its kernel. Since [\varphi] is a surjective homomorphism, the homomorphism theorem states that the factor group [{\bb Z} / n{\bb Z}] is isomorphic to the cyclic group [{\cal C}_n]. The operation in the factor group [{\bb Z} / n{\bb Z}] is `addition modulo n'.








































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