A subgroup order divides the finite group order, with quotient equal to its index.Suggest a slogan
Let \(G\) be a finite group and \(H \leq G\) a subgroup. Then \(\abs H\) divides \(\abs G\), and \([G:H] = \abs G/ \abs H\) [@DF04].
The left cosets of \(H\) partition \(G\) (PR-VUKHO), and each coset \(gH\) has \(\abs H\) elements because \(h\mapsto gh\) is a bijection \(H\to gH\). Hence \(\abs G=[G:H]\,\abs H\).