# 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 ProofTheOrbitOfAnyElementOfAGroupIsASubgroup 2013-03-22 13:30:58 2013-03-22 13:30:58 drini (3) drini (3) 6 drini (3) Proof msc 20A05 Group Subgroup ProofThatEveryGroupOfPrimeOrderIsCyclic ProofOfTheConverseOfLagrangesTheoremForCyclicGroups orbit