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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
\[|G|=|H|\cdot |G/\sim_l|=|H|\cdot |G/\sim_r|.\]
Table of Contents
Proofs: 1
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References
Bibliography
- Ayres jr. Frank: "Theory and Problems of Modern Algebra", McGraw-Hill Book Company Europe, 1978
