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Theorem: Order of Subgroup Divides Order of Finite Group
Let \((G,\ast)\) be a group with a finite order \(G < \infty\) and \(H\subseteq G\) its subgroup. Then the order \(H\) is a divisor of the order \(G\). More precisely, with the equivalence relations defined for the cosets \(a\sim_l b:=a\in bH\) (respectively \(a\sim_r b:=a\in Hb\)) we have
\[G=H\cdot G/\sim_l=H\cdot G/\sim_r.\]
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References
Bibliography
 Ayres jr. Frank: "Theory and Problems of Modern Algebra", McGrawHill Book Company Europe, 1978