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    Lagrange's theorem

    T-SZRXI

    A finite group is partitioned into equal-sized cosets, so subgroup order times index equals group order.Suggest a slogan

    If \(H\) is a subgroup of a finite group \(G\) with index \([G:H]\), then \[ \abs G = [G:H]\abs H. \] In particular, \(\abs H\) divides \(\abs G\), and the order of every element of \(G\) divides \(\abs G\).

    The left cosets of \(H\) partition \(G\). Multiplication by a coset representative is a bijection \(H\to gH\), so every coset has \(\abs H\) elements. There are \([G:H]\) cosets, and summing their equal sizes gives the formula.

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