서브그룹의 인덱스

H ≤ G

Def. Index of H in G.
= the number of left cosets of H in G and denoted by (G : H)
( same as right coset. )

then, since Lagrange's Thm can be rewritten as |G| = |H| (G : H)
the index of H in G can be rewritten as  (G : H) = |G| / |H|

Then, for K ≤ H ≤ G

( G : H ) = |G| / |H|
( H : K ) = |H| / |K|
( G : K ) = |G| / |K|

thus,

( G : K ) = ( G : H ) ( H : K )