# Lagrange's theorem proof

Proof of Lagrange theorem - Order of a subgroup divides order of the group

Lagrange's theorem

How do I know that the cosets of H have the same number of elements?(That equivalence classes form a partition of G.)

Consider a function defined by $f:H\to Ha$ by $f(x)=xa.$ Then it can be easily verified that $f$ is bijection. i.e., $H$ and $Ha$ has same cardinality.
Consider the map $H\to gH$ given by $h\mapsto gh$. Prove that it is bijective.
Let $a$ and $b$ be any two elements, $H$ a finite subgroup. Consider $aH$ and $bH$. If $|aH|\neq|bH|$ then there must be two elements in $H$, say $f$ and $g$ such that either $af=ah$ or $bf=bh$. Assume WLOG $af=ah$. Then $f=h$, a contradiction. Thus all the coasts have the same size.