When H commutes with G, or more generally, when H is normal in G, the index is the size of the quotient group G/H.
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The index of a subgroup H in a group G
is the cardinality of the left or right cosets of H.
This is well defined,
at least for finite groups.
Lagrange's theorem tells us the index of H in G is
|G|/|H|,
whether we use left or right cosets.
Even for infinite groups, the index is well defined.
Invert all the elements of G.
This maps H onto H, and builds a bijection between left and right cosets.
If x is one of our cosreps,
the right coset of x corresponds to the left coset of 1/x, etc.
When H commutes with G, or more generally, when H is normal in G, the index is the size of the quotient group G/H. |
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If K is a subgroup of H, each element of G lies in a coset of H, and after that cosrep is divided out, the remainder lies in a coset of K inside H. Therefore the index of K in G is the index of K in H times the index of H in G.
If H has index 2 in G it is normal. Choose any x not in H and observe that x*H = H*x. After all, there is only one coset outside of H.