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. 6-7
Section 1.1.5. Normal subgroups, factor groups^{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 general, the left and right cosets of a subgroup differ, for example in the symmetry group 3m of an equilateral triangle the left coset decomposition with respect to the subgroup is whereas the right coset decomposition is For particular subgroups, however, it turns out that the left and right cosets coincide, i.e. one has for all . This means that for every and every the element is of the form for some and thus . The element is called the conjugate of by . Note that in the definition of the conjugate element there is a choice whether the inverse element is placed to the left or right of . Depending on the applications that are envisaged and on the preferences of the author, both versions and are found in the literature, but in the context of crystallographic groups it is more convenient to have the inverse to the right of .
An important aspect of conjugate elements is that they share many properties, such as the order or the type of symmetry operation. As a consequence, conjugate symmetry operations have the same type of geometric elements. For example, if is a threefold rotation in three-dimensional space, its geometric element is the line along the rotation axis. The geometric element of a conjugate element is then also a line fixed by a threefold rotation, but in general this line has a different direction.
Definition. A subgroup of is called a normal subgroup if for all and all . This is denoted by . For a normal subgroup , the left and right cosets of with respect to coincide.
Remarks
Examples
For a subgroup of and an element , the conjugates form a subgroup because . This subgroup is called the conjugate subgroup of by . As already noted, conjugation does not alter the type of symmetry operations and their geometric elements, but it is possible that the orientations of the geometric elements are changed.
Using the concept of conjugate subgroups, a normal subgroup is a subgroup that coincides with all its conjugate subgroups . This means that the set of geometric elements of a normal subgroup is not changed by conjugation; the single geometric elements may, however, be permuted by the conjugating element. In the example of the symmetry group 4mm discussed above, the normal subgroup contains the reflections m_{10} and m_{01} with the lines along the coordinate axes as geometric elements. These two lines are interchanged by the fourfold rotation 4^{+}, corresponding to the fact that conjugation by 4^{+} interchanges m_{10} and m_{01}. The concept of conjugation will be discussed in more detail in Section 1.1.8.
One of the main motivations for studying normal subgroups is that they allow us to define a group operation on the cosets of in . The products of any element in the coset with any element in the coset lie in a single coset, namely in the coset . Thus we can define the product of the two cosets and as the coset with representative .
Definition. The set together with the binary operation forms a group, called the factor group or quotient group of by .
The identity element of the factor group is the coset and the inverse element of is the coset .
A familiar example of a factor group is provided by the times on a clock. If it is 8 o'clock (in the morning) now, then we say that in nine hours it will be 5 o'clock (in the afternoon). We regard times as elements of the factor group in which = = . In the factor group , the clock is imagined as a circle of circumference 12 around which the line of integers is wrapped so that integers with a difference of 12 are located at the same position on the circle.
The clock example is a special case of factor groups of the integers. We have already seen that the set of multiples of a natural number n forms a subgroup of index n in . This is a normal subgroup, since is an abelian group. The factor group represents the addition of integers modulo n.
Examples