Index of a subgroup

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically <a href="/facts/Group_theory/NfowcI5H">group theory</a>, the index of a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> H in a group G is the 
number of left <a href="/facts/Coset/zFgQUTLY">cosets</a> of H in G, or equivalently, the number of right cosets of H in G.
The index is denoted 
 
 
 
 
 |
 
 G
 :
 H
 
 |
 
 
 
 {\displaystyle |G:H|}
 
 or 
 
 
 
 [
 G
 :
 H
 ]
 
 
 {\displaystyle [G:H]}
 
 or 
 
 
 
 (
 G
 :
 H
 )
 
 
 {\displaystyle (G:H)}
 
.
Because G is the disjoint union of the left cosets and because each left coset has the same <a href="/facts/Cardinality/m6VPojwT">size</a> as H, the index is related to the <a href="/facts/Order_(group_theory)/Cl3tKGFg">orders</a> of the two groups by the formula

|
        
        G
        
          |
        
        =
        
          |
        
        G
        :
        H
        
          |
        
        
          |
        
        H
        
          |
        
      
    
    {\displaystyle |G|=|G:H||H|}

(interpret the quantities as <a href="/facts/Cardinal_numbers/ccoKwU4R">cardinal numbers</a> if some of them are infinite).
Thus the index 
 
 
 
 
 |
 
 G
 :
 H
 
 |
 
 
 
 {\displaystyle |G:H|}
 
 measures the "relative sizes" of G and H.
For example, let 
 
 
 
 G
 =
 
 Z
 
 
 
 {\displaystyle G=\mathbb {Z} }
 
 be the group of integers under <a href="/facts/Addition/apFWJBFB">addition</a>, and let 
 
 
 
 H
 =
 2
 
 Z
 
 
 
 {\displaystyle H=2\mathbb {Z} }
 
 be the subgroup consisting of the <a href="/facts/Parity_(mathematics)/7zq2UM4V">even integers</a>. Then 
 
 
 
 2
 
 Z
 
 
 
 {\displaystyle 2\mathbb {Z} }
 
 has two cosets in 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
, namely the set of even integers and the set of odd integers, so the index 
 
 
 
 
 |
 
 
 Z
 
 :
 2
 
 Z
 
 
 |
 
 
 
 {\displaystyle |\mathbb {Z} :2\mathbb {Z} |}
 
 is 2. More generally, 
 
 
 
 
 |
 
 
 Z
 
 :
 n
 
 Z
 
 
 |
 
 =
 n
 
 
 {\displaystyle |\mathbb {Z} :n\mathbb {Z} |=n}
 
 for any positive integer n.
When G is <a href="/facts/Finite_group/vv2tvogB">finite</a>, the formula may be written as 
 
 
 
 
 |
 
 G
 :
 H
 
 |
 
 =
 
 |
 
 G
 
 |
 
 
 /
 
 
 |
 
 H
 
 |
 
 
 
 {\displaystyle |G:H|=|G|/|H|}
 
, and it implies
<a href="/facts/Lagrange%27s_theorem_(group_theory)/VgyNjGlz">Lagrange's theorem</a> that 
 
 
 
 
 |
 
 H
 
 |
 
 
 
 {\displaystyle |H|}
 
 divides 
 
 
 
 
 |
 
 G
 
 |
 
 
 
 {\displaystyle |G|}
 
.
When G is infinite, 
 
 
 
 
 |
 
 G
 :
 H
 
 |
 
 
 
 {\displaystyle |G:H|}
 
 is a nonzero <a href="/facts/Cardinal_number/ccoKwU4R">cardinal number</a> that may be finite or infinite.
For example, 
 
 
 
 
 |
 
 
 Z
 
 :
 2
 
 Z
 
 
 |
 
 =
 2
 
 
 {\displaystyle |\mathbb {Z} :2\mathbb {Z} |=2}
 
, but 
 
 
 
 
 |
 
 
 R
 
 :
 
 Z
 
 
 |
 
 
 
 {\displaystyle |\mathbb {R} :\mathbb {Z} |}
 
 is infinite.
If N is a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> of G, then 
 
 
 
 
 |
 
 G
 :
 N
 
 |
 
 
 
 {\displaystyle |G:N|}
 
 is equal to the order of the <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> 
 
 
 
 G
 
 /
 
 N
 
 
 {\displaystyle G/N}
 
, since the underlying set of 
 
 
 
 G
 
 /
 
 N
 
 
 {\displaystyle G/N}
 
 is the set of cosets of N in G.

Index of a subgroup open-in-new

Index of a subgroup