This entry is about the concept in group theory. See at order for the concept in order theory.
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
For a discrete group, its order is the cardinality of the underlying set.
Hence for a finite group, its order is a natural number, the number of its group elements.
For a group and an element, the order of is the smallest positive natural number such that the -fold group product of with itself is the neutral element:
The exponent of a group is the least common multiple of the order of all elements of the group.
Sometimes the term ‘’order’‘ refers to the height of a (group) scheme over a field (of characteristic ) which is defined to be the dimension of the associated ring of functions as a -vector space. Another term for this notion is ‘’rank’’. If this group scheme is moreover p-divisible - which means that is is in fact a codirected diagram of group schemes of order ; in this case is called the order or height of .
For a finite group and an element, there is a close relation between
The element generates a cyclic subgroup . Evidently, the order of the element equals the order of this cyclic subgroup that it generates:
By Lagrange's theorem, if a finite group and a subgroup, then the order of is divisible by the order of . The multiple is called the subgroup index :
Hence with (1) it follows that with any element, the order of divides the order of :
See also
Last revised on August 25, 2024 at 14:25:08. See the history of this page for a list of all contributions to it.