International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A1, ch. 1.5, pp. 31-34
Section 1.5.3. Groups^{a}Abteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany |
The affine group is only one example of the more general concept of a group. The following axiomatic definition sometimes makes it easier to examine general properties of groups.
Definition 1.5.3.1.1. A group is a set with a mapping , called the composition law or multiplication of , satisfying the following three axioms:
Normally the symbol · is omitted, hence the product is just written as and the set is called a group.
One should note that in particular property (i), the associative law, of a group is something very natural if one thinks of group elements as mappings. Clearly the composition of mappings is associative. In general, one can think of groups as groups of mappings as explained in Section 1.5.3.2.
A subset of elements of a group which themselves form a group is called a subgroup:
The affine group is an example of a group where is given by the composition of mappings. The unit element is the identity mapping given by the matrix which also represents the translation by the vector . The composition of two affine mappings is again an affine mapping and the inverse of an affine mapping has matrix Since the inverse of an isometry and the composition of two isometries are again isometries, the set of isometries is a subgroup of the affine group . The translation subgroup is a subgroup of .
Any vector space is a group with the usual vector addition as composition law. Therefore is also a group.
Remarks
Example 1.5.3.1.3
A well known group is the addition group of integers where · is normally denoted by + and the unit element is 0. The group is generated by . Other generating sets are for example or . Taking two integers which are divisible by some fixed integer , then the sum and the addition inverses and are again divisible by p. Hence the set of all integers divisible by p is a subgroup of . It is generated by .
Most of the groups in crystallography, for example , , , have infinite order.
Groups that are generated by one element are called cyclic. The cyclic group of order n is called . (We prefer to use three letters to denote the mathematical names of frequently occurring groups, since the more common symbol could possibly cause confusion with the Schoenflies symbol .)
The group is not generated by a finite set.
These two groups and have the property that for all elements and in the group it holds that . Hence these two groups are Abelian in the sense of the following:
The affine group is defined via its action on the affine space . In general, the greatest significance of groups is that they act on sets.
Definition 1.5.3.2.1. Let be a group. A non-empty set M is called a (left) -set if there is a mapping satisfying the following conditions:
If M is a -set, one also says that acts on M.
Example 1.5.3.2.2
Definition 1.5.3.2.3. Let be a group and M a -set. If , then the set is called the orbit of m under .
The -set M is called transitive if for any consists of a single orbit under .
If then the stabilizer of m in is .
The kernel of the action of on M is the intersection of the stabilizers of all elements in M, M is called a faithful -set and the action of on M is also called faithful if the kernel of the action is trivial, .
Remarks
Example 1.5.3.2.4 (Example 1.5.3.2.2 cont.)
We now introduce some terminology for groups which is nicely formulated using -sets.
Definition 1.5.3.2.5. The orbit of under the action of the subgroup is the right coset (cf. IT A, Section 8.1.5 ). Analogously one defines a left coset as and denotes the set of left cosets by .
If the number of left cosets (which is always equal to the number of right cosets) of in is finite, then this number is called the index of in . If this number is infinite, one says that the index of in is infinite.
Example 1.5.3.2.6.
is a coset of in , namely If one thinks of as an infinite sheet of paper and puts uncountably many such sheets of paper (one for each real number) one onto the other, one gets the whole -space .
Remark. The set of left cosets is a left -set with the operation for all . The kernel of the action is the intersection of all subgroups of that are conjugate to and is called the core of : .
We now assume that is finite. Let be a subgroup of . Then the set is partitioned into left cosets of , , where is the index of in . Since the orders of the left cosets of are all equal to the order of , one gets
Theorem 1.5.3.2.7. (Theorem of Lagrange.) Let be a subgroup of the finite group . Then In particular, the order of any subgroup of and also the index of any subgroup of are divisors of the group order .
The -set is only a special case of a -set. It is a transitive -set. If is a transitive -set, then the mapping , is a bijection (in fact an isomorphism of -sets in the sense of Definition 1.5.3.4.1 below). Therefore the number of elements of M, which is the length of the orbit of m under , equals the index of the stabilizer of m in , whence one gets the following generalization of the theorem of Lagrange:
Up to now, we have only considered the action of upon via multiplication. There is another natural action of on itself via conjugation: defined by for all group elements and elements m in the -set . The stabilizer of m is called the centralizer of m in , If is a set of group elements, then the centralizer of M is the intersection of the centralizers of the elements in M:
Definition 1.5.3.2.9. also acts on the set of all subgroups of by conjugation, . The stabilizer of an element is called the normalizer of and denoted by . is called a normal subgroup of (denoted as ) if .
Remarks
Normal subgroups play an important role in the investigation of groups. If is a normal subgroup, then the left coset equals the right coset for all , because .
The most important property of normal subgroups is that the set of left cosets of in forms a group, called the factor group , as follows: The set of all products of elements of two left cosets of again forms a left coset of . Let . Then This defines a natural product on the set of left cosets of in which turns this set into a group. The unit element is .
Hence the philosophy of normal subgroups is that they cut the group into pieces, where the two pieces and are again groups.
Example 1.5.3.2.10. The group is Abelian. For any number , the set is a subgroup of . Hence is a normal subgroup of . The factor group inherits the multiplication from the multiplication in , since for all . If p is a prime number, then all elements in have a multiplicative inverse, and therefore is a field, the field with p elements.
Proof. Let , . Then , where , since is a subgroup of , and , since is a normal subgroup of . QED
A nice application of the notion of -sets are the three theorems of Sylow. By Theorem 1.5.3.2.7, the order of any subgroup of a group divides the order of . But conversely, given a divisor d of , one cannot predict the existence of a subgroup of with . If is a prime power that divides , then the following theorem says that such a subgroup exists.
Theorem 1.5.3.3.1. (Sylow) Let be a finite group and p be a prime such that divides the order of . Then possesses m subgroups of order , where satisfies .
Theorem 1.5.3.3.2. (Sylow) If for some prime p not dividing s, then all subgroups of order of are conjugate in . Such a subgroup of order is called a Sylow p-subgroup.
Combining these two theorems with Theorem 1.5.3.2.8, one gets Sylow's third theorem:
Proofs of the three theorems above can be found in Ledermann (1976), pp. 158–164.
If one wants to compare objects such as groups or -sets, to be able to say when they should be considered as equal, the concept of isomorphisms should be used:
Definition 1.5.3.4.1. Let and be groups and M and N be -sets.
Example 1.5.3.4.2. In Example 1.5.3.1.3, the group homomorphism defined by is a group isomorphism (from the group onto its subgroup ).
Example 1.5.3.4.3. For any group element , conjugation by defines an automorphism of . In particular, if is a subgroup of , then and its conjugate subgroup are isomorphic.
Philosophy: If and are isomorphic groups, then all group-theoretical properties of and are the same. The calculations in can be translated by the isomorphism to calculations in . Sometimes it is easier to calculate in one group than in the other and translate the result back via the inverse of the isomorphism. For example, the isomorphism between and in Section 1.5.2 is an isomorphism of groups. It even respects scalar multiplication with real numbers, so in fact it is an isomorphism of vector spaces. While the composition of translations is more concrete and easier to imagine, the calculation of the resulting vector is much easier in . The concept of isomorphism says that you can translate to the more convenient group for your calculations and translate back afterwards without losing anything.
Note that a homomorphism is injective, i.e. is an isomorphism onto its image, if and only if its kernel is trivial .
Example 1.5.3.4.4
The mapping is a homomorphism of the group into . The kernel of this homomorphism is and the image of the mapping is the translation subgroup of . Hence the groups and are isomorphic.
The affine group acts (as group of group automorphisms) on the normal subgroup via conjugation: . We have seen already in Example 1.5.3.2.4 (b) that it also acts (as a group of linear mappings) on . The mapping is an isomorphism of -sets.
[cf. Ledermann (1976), pp. 68–73.]
Remark. If is a homomorphism and is a normal subgroup of , then the pre-image is a normal subgroup of . In particular, it holds that .
Hence the factor group is a well defined group. The following theorem says that this group is isomorphic to the image of :
Theorem 1.5.3.5.1. (First isomorphism theorem.) Let be a homomorphism of groups. Then is an isomorphism between the factor group and the image group of , which is a subgroup of .
Theorem 1.5.3.5.2. (Third isomorphism theorem.) Let be a normal subgroup of the group and be an arbitrary subgroup of . Then is a normal subgroup of and (For the definition of the group see Proposition 1.5.3.2.11.)
Remarks
Let us consider the tetrahedral group, Schoenflies symbol , which is defined as the symmetry group of a tetrahedron. It permutes the four apices of the tetrahedron and hence every element of defines a bijection of onto itself. The only element that fixes all the apices is . Therefore the set V is a faithful -set. Let us calculate the order of . Since there are elements in that map the first apex onto each one of the other apices, V is a transitive -set. Let be the stabilizer of . By Theorem 1.5.3.2.8, . The group is generated by the threefold rotation around the `diagonal' of the tetrahedron through and the reflection at the symmetry plane of the tetrahedron which contains the edge . In particular, acts transitively on the set . The stabilizer of in is the cyclic group generated by . (The Schoenflies notation for is and the Hermann–Mauguin symbol is m.) Therefore and . In fact, we have seen that is isomorphic to the group of all bijections of V onto itself, which is the symmetric group of degree 4 and the group is the symmetric group on . The Schoenflies notation for is and its Hermann–Mauguin symbol is .
In general, let be a natural number. Then the group of all bijective mappings of the set onto itself is called the symmetric group of degree n and denoted by The alternating group is the normal subgroup consisting of all even permutations of .
Let us construct a normal subgroup of . The tetrahedral group contains three twofold rotations around the three axes of the tetrahedron through the midpoints of opposite edges. Since permutes these three axes and hence conjugates the three rotations into each other, the group generated by these three rotations is a normal subgroup of . Since these three rotations commute with each other, the group is Abelian. Now and hence (in Schoenflies notation) (Hermann–Mauguin symbol) is of order 4. There are three normal subgroups of order 2 in , namely for . The factor group is again of order 2. Since all groups of order 2 are cyclic, . The set is the set of all products of elements from the two normal subgroups and , hence is isomorphic to the direct product in the sense of the following definition.
Definition 1.5.3.6.1. [cf. Ledermann (1976), Section 13.] Let and be two groups. Then the direct product is the group with multiplication .
Let us return to the example above. The centralizer of one of the three rotations, say of , is of index 3 in and hence a Sylow 2-subgroup of with order 8. Following Schoenflies, we will denote this group by (another Schoenflies symbol for this group is and its Hermann–Mauguin symbol is ).
The group above is contained in . It is its own centralizer in : . Therefore the factor group acts faithfully (and transitively) on the set . The stabilizer of is the subgroup constructed above. Using this, one easily sees that .
Another normal subgroup in is the set of all rotations in . This group contains the normal subgroup above of index 3 and is of index 2 in (and hence has order 12). It is isomorphic to , the alternating group of degree 4, and has Schoenflies symbol T and Hermann–Mauguin symbol .
References
Ledermann, W. (1976). Introduction to group theory. London: Longman. (German: Einführung in die Gruppentheorie, Braunschweig: Vieweg, 1977.)