Editing Index of a subgroup (section)

==Properties==
* If ''H'' is a subgroup of ''G'' and ''K'' is a subgroup of ''H'', then
::<math>|G:K| = |G:H|\,|H:K|.</math>
* If ''H'' and ''K'' are subgroups of ''G'', then
::<math>|G:H\cap K| \le |G : H|\,|G : K|,</math>
:with equality if <math>HK=G</math>.  (If <math>|G:H\cap K|</math> is finite, then equality holds if and only if <math>HK=G</math>.)
* Equivalently, if ''H'' and ''K'' are subgroups of ''G'', then
::<math>|H:H\cap K| \le |G:K|,</math>
:with equality if <math>HK=G</math>.  (If <math>|H:H\cap K|</math> is finite, then equality holds if and only if <math>HK=G</math>.)
* If ''G'' and ''H'' are groups and <math>\varphi \colon G\to H</math> is a [[homomorphism]], then the index of the [[kernel (algebra)|kernel]] of <math>\varphi</math> in ''G'' is equal to the order of the image:
::<math>|G:\operatorname{ker}\;\varphi|=|\operatorname{im}\;\varphi|.</math>
* Let ''G'' be a group [[Group action (mathematics)|acting]] on a [[set (mathematics)|set]] ''X'', and let ''x''&nbsp;∈&nbsp;''X''.  Then the [[cardinality]] of the [[orbit (group theory)|orbit]] of ''x'' under ''G'' is equal to the index of the [[stabilizer subgroup|stabilizer]] of ''x'':
::<math>|Gx| = |G:G_x|.\!</math>
:This is known as the [[orbit-stabilizer theorem]].
* As a special case of the orbit-stabilizer theorem, the number of [[conjugacy class|conjugates]] <math>gxg^{-1}</math> of an element <math>x \in G</math> is equal to the index of the [[centralizer]] of ''x'' in ''G''.
* Similarly, the number of conjugates <math>gHg^{-1}</math> 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 [[core (group)|normal core]] of ''H'' satisfies the following inequality:
::<math>|G:\operatorname{Core}(H)| \le |G:H|!</math>
:where ! denotes the [[factorial]] function; this is discussed further [[#Finite index|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., it 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]].