Index of a subgroup: Difference between revisions

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Definition

Symbol-free definition

The index of a subgroup in a group is the cardinality of the coset space of the subgroup.

When the group is finite, then by Lagrange's theorem, the index of a subgroup is the ratio of the order of the group to the order of the subgroup.

Definition with symbols

Given a subgroup $H$ of a group $G$ , the index of $H$ in $G$ is the cardinality of the coset space $G / H$ , that is, the number of left (respectively right) cosets of $H$ in $G$ .

If $G$ is a finite group, then the index of $H$ in $G$ is the ratio of the cardinality of $G$ to the cardinality of $H$ .

The index is denoted as $[G : H]$ .

Facts

Multiplicativity of the index

Further information: Index is multiplicative If $H \leq K \leq G$ , then we have:

$[G : K] [K : H] = [G : H]$

In other words, the number of cosets of $H$ in $G$ equals the number of cosets of $H$ in $K$ , times the number of cosets of $K$ in $G$ .

In fact, more is true. We can set up a bijection as follows:

$G / K \times K / H \to G / H$

However, this bijection is not a natural one, and, in order to define it, we first need to choose a system of coset representatives of $H$ .

Effect of intersection on the index

Further information: Conjugate-intersection index theorem If $H_{1}$ and $H_{2}$ are two subgroups of $G$ , then the index of $H_{1} \cap H_{2}$ is bounded above by the product of the indices of $H_{1}$ and of $H_{2}$ .

This follows as a consequence of the product formula. Note that equality holds if and only if $H_{1} H_{2} = G$ .

Note that in case $H_{1}$ and $H_{2}$ are conjugate subgroups of index $r$ , the index of $H_{1} \cap H_{2}$ is bounded above by $r (r - 1)$ .

Related notions

For double cosets and multicosets

Related subgroup properties

@@ Line 20: / Line 20: @@
 ===Multiplicativity of the index===
+{{further|[[Index is multiplicative]]}}
 If <math>H \le K \le G</math>, then we have:
@@ Line 34: / Line 34: @@
 ===Effect of intersection on the index===
+{{further|[[Conjugate-intersection index theorem]]}}
 If <math>H_1</math> and <math>H_2</math> are two subgroups of <math>G</math>, then the index of <math>H_1 \cap H_2</math> is bounded above by the product of the indices of <math>H_1</math> and of <math>H_2</math>.
 This follows as a consequence of the [[product formula]]. Note that equality holds if and only if <math>H_1H_2 = G</math>.
-Note that in case <math>H_1</math> and <math>H_2</math> are conjugate subgroups of index <math>r</math>, the index of <math>H_1 \cap H_2</math> is bounded above by <math>r(r-1)</math>. {{further|[[Conjugate-intersection index theorem]]}}
+Note that in case <math>H_1</math> and <math>H_2</math> are conjugate subgroups of index <math>r</math>, the index of <math>H_1 \cap H_2</math> is bounded above by <math>r(r-1)</math>.
 ==Related notions==