# Category:Order of Group Elements

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This category contains results about Order of Group Elements.

Definitions specific to this category can be found in Definitions/Order of Group Elements.

The **order of $x$ (in $G$)**, denoted $\order x$, is the smallest $k \in \Z_{> 0}$ such that $x^k = e_G$.

## Subcategories

This category has the following 8 subcategories, out of 8 total.

### E

## Pages in category "Order of Group Elements"

The following 41 pages are in this category, out of 41 total.

### E

- Element of Finite Group is of Finite Order
- Element to Power of Multiple of Order is Identity
- Elements of Abelian Group whose Order Divides n is Subgroup
- Equal Order Elements may not be Conjugate
- Equal Powers of Finite Order Element
- Equal Powers of Group Element implies Finite Order
- Equivalence of Definitions of Finite Order Element
- Equivalence of Definitions of Infinite Order Element
- Equivalence of Definitions of Order of Group Element
- Even Order Group has Odd Number of Order 2 Elements
- Even Order Group has Order 2 Element

### G

### N

### O

- Odd Order Group Element is Square
- Order of Conjugate Element equals Order of Element
- Order of Conjugate Element equals Order of Element/Corollary
- Order of Cyclic Group equals Order of Generator
- Order of Element divides Order of Centralizer
- Order of Element Divides Order of Finite Group
- Order of Element in Group equals its Order in Subgroup
- Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2
- Order of Group Element equals Order of Coprime Power
- Order of Group Element equals Order of Coprime Power/Corollary
- Order of Group Element equals Order of Inverse
- Order of Group Element not less than Order of Power
- Order of Group Product
- Order of Power of Group Element
- Order of Product of Commuting Group Elements of Coprime Order is Product of Orders