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Concept
Index of a subgroup
Summary
In mathematics, specifically group theory, the index of a subgroup H in a group G is the
number of left cosets of H in G, or equivalently, the number of right cosets of H in G.
The index is denoted or or .
Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula
(interpret the quantities as cardinal numbers if some of them are infinite).
Thus the index measures the "relative sizes" of G and H.
For example, let be the group of integers under addition, and let be the subgroup consisting of the even integers. Then has two cosets in , namely the set of even integers and the set of odd integers, so the index is 2. More generally, for any positive integer n.
When G is finite, the formula may be written as , and it implies
Lagrange's theorem that divides .
When G is infinite, is a nonzero cardinal number that may be finite or infinite.
For example, , but is infinite.
If N is a normal subgroup of G, then is equal to the order of the quotient group , since the underlying set of is the set of cosets of N in G.
If H is a subgroup of G and K is a subgroup of H, then
If H and K are subgroups of G, then
with equality if . (If is finite, then equality holds if and only if .)
Equivalently, if H and K are subgroups of G, then
with equality if . (If is finite, then equality holds if and only if .)
If G and H are groups and is a homomorphism, then the index of the kernel of in G is equal to the order of the image:
Let G be a group acting on a set X, and let x ∈ X. Then the cardinality of the orbit of x under G is equal to the index of the stabilizer of x:
This is known as the orbit-stabilizer theorem.
As a special case of the orbit-stabilizer theorem, the number of conjugates of an element is equal to the index of the centralizer of x in G.
Similarly, the number of conjugates of a subgroup H in G is equal to the index of the normalizer of H in G.
Official source
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