Order of a group

Order of a group
Tag	`0SEX9F9J5Z`
URI	`m2://0SEX9F9J5Z`
Kind	definition
Writing	W1 stated
MSC	20A05
Defines	Order of a group
Uses	Group

Markdown · JSON · Edit

Definition

Let $G$ be a group. The order of $G$ , written $|G|$ , is the cardinality of the underlying set of $G$ . When $G$ is finite, $|G|$ is the number of elements of $G$ . When $G$ is infinite, $G$ has infinite order.

Remarks

The trivial group has order $1$ , and the group of $n$ th roots of unity has order $n$ . The group $(\mathbb{Z}, +)$ has infinite order. Lagrange's theorem says that the order of a subgroup of a finite group divides the order of the group.

Explanations

L1 human

The order of a group tells you how big the group is. For example, the rotations of a square that map the square to itself form a group of order $4$ . The word "order" has a second meaning in group theory: the order of an element.

What this node needs for W2

edge instance_of in: one example (0 of 1)
edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 2 nodes in this order. The foundations come first, and the list ends with this node.

Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.

no evidence level

Group concept
Order of a group definition

What depends on this

30 nodes use this node, directly or through other nodes.

A finite group is nilpotent iff it is the product of its Sylow subgroups 00Y9N80VSS
Order of a cyclic subgroup equals the order of its generator 031M8HX7DA
Order of a group is the index times the order of the subgroup 04DC0SRN2K
Groups of prime order are cyclic 05MM4EK4SA
Cyclic groups of the same order are isomorphic 0736AXH2ME
Classification of finite abelian groups 07CQR9BW3R
Product of cyclic groups is cyclic iff the orders are coprime 08FDFEF8D1
Order of an element divides the order of the group 08V7JHS3QZ
Fixed points of a p-group action 098D2E7DEH
Second Sylow theorem 09GCQ5STVJ
The number of Sylow subgroups is the index of the normaliser 09ZJ6CAK7S
Order of the dihedral group 0BZVNWJ352
Cyclic group of order n 0G0B3DRG65
Abelian simple groups have prime order 0JJV80RXHD
Cauchy's theorem 0JM7PA288B
Order of the general linear group over a finite field 0JQ1KNQW2K
The number of Sylow subgroups divides the index 0M3PYJ3TSE
A finite abelian group is cyclic iff its exponent equals its order 0MC0AQZSHE
Order of the symmetric group 0PCKBBMEXC
Third Sylow theorem 0Q8DGAZSJ4
The group of units of Z/pZ is cyclic 0QTHB6PE52
Frattini argument 0QTZDKJEDV
Structure theorem for finitely generated abelian groups 0RY0B72XAS
Size of an orbit times the size of the stabiliser 0S19CWFZM2
Order of the group of units of Z/nZ is Euler's totient 0S7495H4HM
Order of the alternating group 0SZ1KMNWVE
First Sylow theorem 0V98WY94B8
Groups of order pq 0YBR6QPY4Y
Burnside's lemma 0Z47M1B2VS
g to the order of the group is the identity 0ZEPNS9BXK

Sources

Web https://en.wikipedia.org/wiki/Order_(group_theory), lead section

What links here

Used by

Order of a cyclic subgroup equals the order of its generator 031M8HX7DA
Order of a group is the index times the order of the subgroup 04DC0SRN2K
Cyclic groups of the same order are isomorphic 0736AXH2ME
Order of the dihedral group 0BZVNWJ352
Cyclic group of order n 0G0B3DRG65
Order of the general linear group over a finite field 0JQ1KNQW2K
Order of the symmetric group 0PCKBBMEXC
Order of the group of units of Z/nZ is Euler's totient 0S7495H4HM