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YCor
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Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we conclude that $\alpha$ has order $p$ in the orbifold fundamental group $\pi_1^{orb}(M)$$\pi_1^{\mathrm{orb}}(M)$?

Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we conclude that $\alpha$ has order $p$ in the orbifold fundamental group $\pi_1^{orb}(M)$?

Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we conclude that $\alpha$ has order $p$ in the orbifold fundamental group $\pi_1^{\mathrm{orb}}(M)$?

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Order of a loop around a cone point

Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we conclude that $\alpha$ has order $p$ in the orbifold fundamental group $\pi_1^{orb}(M)$?