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Tour:Order of a group

This article adapts material from the main article: order of a group

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WHAT YOU NEED TO DO: Go through the definition below and understand it.

Definition

QUICK PHRASES: size of a group, cardinality of a group, size of the underlying set, number of elements

Symbol-free definition

The order of a group is the cardinality (i.e., size, or number of elements) of its underlying set.

Definition with symbols

The order of a group G is the cardinality (i.e., size, or number of elements) of G as a set. it is denoted as \left| G \right|.

Note that a finite group is a group whose underlying set is finite, i.e., the size of the underlying set is finite. The order of a finite group is thus a natural number (note that the order cannot be zero because every group contains the identity element and is hence nonempty). For an infinite group, the order is an infinite cardinal.

Examples

  • The trivial group, which is the group with only the identity element, has order 1. In fact, it is the only group (up to isomorphism}) that has order 1.


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