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. 4-5
Section 1.1.3. Subgroups^{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 |
The group of symmetry operations of a crystal pattern may alter if the crystal undergoes a phase transition. Often, some symmetries are preserved, while others are lost, i.e. symmetry breaking takes place. The symmetry operations that are preserved form a subset of the original symmetry group which is itself a group. This gives rise to the concept of a subgroup.
Definition. A subset is called a subgroup of if its elements form a group by themselves. This is denoted by . If is a subgroup of , then is called a supergroup of . In order to be a subgroup, is required to contain the identity element of , to contain inverse elements and to be closed with respect to composition of elements. Thus, technically, every group is a subgroup of itself.
The subgroups of that are not equal to are called proper subgroups of . A proper subgroup of is called a maximal subgroup if it is not a proper subgroup of any proper subgroup of .
It is often convenient to specify a subgroup of by a set of generators. This is denoted by . The order of is not a priori obvious from the set of generators. For example, in the symmetry group 4mm of the square the pairs and both generate subgroups of order 4, whereas the pair generates the full group of order 8.
The subgroups of a group can be visualized in a subgroup diagram. In such a diagram the subgroups are arranged with subgroups of higher order above subgroups of lower order. Two subgroups are connected by a line if one is a maximal subgroup of the other. By following downward paths in this diagram, all group–subgroup relations in a group can be derived. Additional information is provided by connecting subgroups of the same order by a horizontal line if they are conjugate (see Section 1.1.7).
Examples