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Index of a subgroup ：ウィキペディア英語版

Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the "relative size" of ''H'' in ''G'': equivalently, the number of "copies" (cosets) of ''H'' that fill up ''G''. For example, if ''H'' has index 2 in ''G'', then intuitively "half" of the elements of ''G'' lie in ''H''. The index of ''H'' in ''G'' is usually denoted |''G'' : ''H''| or () or (''G'':''H'').
Formally, the index of ''H'' in ''G'' is defined as the number of cosets of ''H'' in ''G''. (The number of left cosets of ''H'' in ''G'' is always equal to the number of right cosets.) For example, let Z be the group of integers under addition, and let 2Z be the subgroup of Z consisting of the even integers. Then 2Z has two cosets in Z (namely the even integers and the odd integers), so the index of 2Z in Z is two. To generalize,
:

|\mathbf:n\mathbf| = n

for any positive integer ''n''.
If ''N'' is a normal subgroup of ''G'', then the index of ''N'' in ''G'' is also equal to the order of the quotient group ''G'' / ''N'', since this is defined in terms of a group structure on the set of cosets of ''N'' in ''G''.
If ''G'' is infinite, the index of a subgroup ''H'' will in general be a non-zero cardinal number. It may be finite - that is, a positive integer - as the example above shows.
If ''G'' and ''H'' are finite groups, then the index of ''H'' in ''G'' is equal to the quotient of the orders of the two groups:
:

|G:H| = \frac.

This is Lagrange's theorem, and in this case the quotient is necessarily a positive integer.
==Properties==

* If ''H'' is a subgroup of ''G'' and ''K'' is a subgroup of ''H'', then
::

|G:K| = |G:H|\,|H:K|.

* If ''H'' and ''K'' are subgroups of ''G'', then
::

|G:H\cap K| \le |G : H|\,|G : K|,

: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
::

|H:H\cap K| \le |G:K|,

: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:
::

|G:\operatorname\;\varphi|=|\operatorname\;\varphi|.

* 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'':
::

|Gx| = |G:G_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:
::

|G:\operatorname(H)| \le |G:H|!

: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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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