Index of A Subgroup - Properties

Properties

• 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 HK = G. (If |G : H ∩ K| is finite, then equality holds if and only if HK = G.)

• Equivalently, if H and K are subgroups of G, then

with equality if HK = G. (If |H : H ∩ K| is finite, then equality holds if and only if HK = G.)

• If G and H are groups and φ: G → H is a homomorphism, then the index of the kernel of φ 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 gxg−1 of an element x ∈ G is equal to the index of the centralizer of x in G.
• Similarly, the number of conjugates gHg−1 of a subgroup H in G is equal to the index of the normalizer of H in G.
• If H is a subgroup of G, the index of the normal core of H satisfies the following inequality:

where ! denotes the factorial function; this is discussed further below.

• As a corollary, if the index of H in G is 2, or for a finite group the lowest prime p that divides the order of G, then H is normal, as the index of its core must also be p, and thus H equals its core, i.e., is normal.
• Note that a subgroup of lowest prime index may not exist, such as in any simple group of non-prime order, or more generally any perfect group.