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order (of a group) (Definition)

The order of a group $G$ is the number of elements of $G$ denoted $|G|$ if $|G|$ is finite, then $G$ is said to be a finite group.

The order of an element $g \in G$ is the smallest positive integer $n$ such that $g^n=e$ where $e$ is the identity element; if there is no such $n$ then $g$ is said to be of infinite order. By Lagrange's theorem, the order of any element in a finite group divides the order of the group.




"order (of a group)" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: group, cardinality, orders of elements in integral domain, order (of a ring), ideal of elements with finite order

Other names:  order
Also defines:  finite group, infinite order, order (of a group element)

Attachments:
order of elements in finite groups (Theorem) by rm50
a group of even order contains an element of order 2 (Theorem) by azdbacks4234
non-isomorphic groups of given order (Theorem) by pahio
order of products (Theorem) by pahio
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Cross-references: divides, Lagrange's theorem, identity element, integer, positive, finite, number, group
There are 201 references to this entry.

This is version 6 of order (of a group), born on 2002-04-23, modified 2005-02-10.
Object id is 2871, canonical name is OrderGroup.
Accessed 24731 times total.

Classification:
AMS MSC20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)

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