stylistic:style_transfer, structural:summarization, information:extraction, information:classification
Proof

The left cosets of H in G are the equivalence classes of a certain equivalence relation on G: specifically, call x and y in G equivalent if there exists h in H such that x = yh. Therefore, the left cosets form a partition of G.Each left coset aH has the same cardinality as H because x ↦ a x {\displaystyle x\mapsto ax} defines a bijection H → a H {\displaystyle H\to aH} (the inverse is y ↦ a − 1 y {\displaystyle y\mapsto a^{-1}y}).The number of left cosets is the index.By the previous three sentences,