Lagrange's Divisibility Theorem of Order of Subgroups

Theorem

Suppose $G$ is a finite Group and $H \leq G$ (see Subgroup). Then $| H |$ divides $| G |$ (see Order (Groups)).

Proof

The left cosets of $H$ partition $G$ , so they all have size $| H |$ and are disjoint. Thus:

| G | = (# cosets of H in G) | H |

Thus $| H |$ divides $| G |$ .

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An alternative method of this is saying:

Theorem

The number of left cosets of $H$ in $G$ equals $\frac{| G |}{| H |}$

Proof

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