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2 edited tags; added 30 characters in body

Let me first state the definitions :

A not-nullhomotopic closed curve / loop $c$ on a an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ;$ A closed curve / loop $c$ is called primitive if in the fundamental group $\pi_1(X,c(1)),$ the homotopy class $[c]$ can NOT be written as $[c]= [\gamma]^n$ for some closed curve $\gamma$ with $\gamma(0)=\gamma(1)=c(0)=c(1)$ and for some $n\ge 2$.

My question is : rigorously prove that simple closed curves are primitive .

It is visually pretty clear , but I have difficulty proving it. Thanks !

1

# How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

Let me first state the definitions :

A closed curve / loop $c$ on a surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ;$ A closed curve / loop $c$ is called primitive if in the fundamental group $\pi_1(X,c(1)),$ the homotopy class $[c]$ can NOT be written as $[c]= [\gamma]^n$ for some closed curve $\gamma$ with $\gamma(0)=\gamma(1)=c(0)=c(1)$ and for some $n\ge 2$.

My question is : rigorously prove that simple closed curves are primitive .

It is visually pretty clear , but I have difficulty proving it. Thanks !