# Group theory

# Group theory

A group is a set of elements, such as symmetry operations, which fulfil certain group axioms. The composition of group elements yields again a group element. The results of the compositions can be presented in a group multiplication table. A subset of the group elements which again fulfils the group axioms is a subgroup. If there exists no intermediate group between a group and its subgroup, the subgroup is a maximal subgroup. Two groups are isomorphic if they have the same group multiplication table. Following certain rules, a group can be decomposed to cosets with respect to a subgroup. The number of cosets is the index of the subgroup in the group. Two subgroups are conjugate if they are transformed the one to the other by an element of the group. A normal subgroup is transformed onto itself by all elements of the group. If the cosets of a group with respect to a normal subgroup are considered to be new group elements, they make up a new group, a factor group. The action of a (symmetry) group on a set (of, e.g., atoms) deals with the consequences for this set. A crystallographic point orbit is the set of all points that are symmetry equivalent by the action of a space group.

*Keywords:*
group, group element, group axioms, subgroup, multiplication table, cosets, index, normal subgroup, factor group, crystallographic point orbit

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