PlanetMath: order (of a group)

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order (of a group)

(Definition)

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

The order of an element $g \in G$ is the smallest positive integer such that , where is the identity element; if there is no such , then 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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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

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Cross-references: divides, Lagrange's theorem, identity element, integer, positive, finite, group
There are 169 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 19075 times total.

Classification:

AMS MSC:

20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)

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