Proof: The orbit of any element of a group is a subgroup

Following is a proof that, if $G$ is a group and $g\in G$ , then $\langle g\rangle\leq G$ . Here $\langle g\rangle$ is the orbit of $g$ and is defined as

$\langle g\rangle=\{g^{n}:n\in\mathbbmss{Z}\}$

Since $g\in\langle g\rangle$ , then $\langle g\rangle$ is nonempty.

Let $a,b\in\langle g\rangle$ . Then there exist $x,y\in{\mathbbmss{Z}}$ such that $a=g^{x}$ and $b=g^{y}$ . Since $ab^{-1}=g^{x}(g^{y})^{-1}=g^{x}g^{-y}=g^{x-y}\in\langle g\rangle$ , it follows that $\langle g\rangle\leq G$ .

Title	Proof: The orbit of any element of a group is a subgroup
Canonical name	ProofTheOrbitOfAnyElementOfAGroupIsASubgroup
Date of creation	2013-03-22 13:30:58
Last modified on	2013-03-22 13:30:58
Owner	drini (3)
Last modified by	drini (3)
Numerical id	6
Author	drini (3)
Entry type	Proof
Classification	msc 20A05
Related topic	Group
Related topic	Subgroup
Related topic	ProofThatEveryGroupOfPrimeOrderIsCyclic
Related topic	ProofOfTheConverseOfLagrangesTheoremForCyclicGroups
Defines	orbit