Quotient Group is Group

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Theorem

Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.


Then the quotient group $G / N$ is indeed a group.


Corollary

Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.


If $G$ is finite, then:

$\index G N = \order {G / N}$


Proof

By Subgroup is Normal iff Left Cosets are Right Cosets, the set of left cosets for $N$ equals the set of right cosets.

It follows that $G / N$ does not depend on whether left cosets are used to define it or right cosets.

Without loss of generality, we will work with the left cosets.


By definition of quotient group, the elements of $G / N$ are the cosets of $N$ in $G$, where the group operation is defined as:

$\paren {a N} \paren {b N} = \paren {a b} N$


The operation has been shown in Coset Product is Well-Defined to be a well-defined operation.

Now we need to demonstrate that $G / N$ is a group.


Group Axiom $\text G 0$: Closure

This follows from Coset Product is Well-Defined.

As $a b \in G$, it follows that $\paren {a b} N$ is a left coset.

Thus $G / N$ is closed.

$\Box$


Group Axiom $\text G 1$: Associativity

The associativity of coset product follows directly from Subset Product within Semigroup is Associative:

\(\ds a N \paren {b N c N}\) \(=\) \(\ds a N \paren {b c N}\) Definition of Coset Product
\(\ds \) \(=\) \(\ds a \paren {b c} N\) Definition of Coset Product
\(\ds \) \(=\) \(\ds \paren {a b} c N\) Group Axiom $\text G 1$: Associativity for $G$
\(\ds \) \(=\) \(\ds \paren {a b} N c N\) Definition of Coset Product
\(\ds \) \(=\) \(\ds \paren {a N b N} c N\) Definition of Coset Product

Thus $G / N$ is associative.

$\Box$


Group Axiom $\text G 2$: Existence of Identity Element

The left coset $e N = N$ serves as the identity:

\(\ds \forall x \in G: \, \) \(\ds N \paren {x N}\) \(=\) \(\ds \paren {e N} \paren {x N}\) Coset by Identity
\(\ds \) \(=\) \(\ds \paren {e x} N\) Definition of Coset Product
\(\ds \) \(=\) \(\ds x N\) Group Axiom $\text G 2$: Existence of Identity Element for $G$


Similarly $\paren {x N} N = x N$.

$\Box$


Group Axiom $\text G 3$: Existence of Inverse Element

We have $\paren {x N}^{-1} = x^{-1} N$:

\(\ds \paren {x N} \paren {x^{-1} N}\) \(=\) \(\ds \paren {x x^{-1} } N\) Definition of Coset Product
\(\ds \) \(=\) \(\ds e N\) Definition of Inverse Element for $G$
\(\ds \) \(=\) \(\ds N\) Group Axiom $\text G 2$: Existence of Identity Element for $G$


Similarly $\paren {x^{-1} N} \paren {x N} = N$.


Thus $x^{-1} N$ is the inverse of $x N$.

$\Box$


All the group axioms are seen to be fulfilled, and $G / N$ has been shown to be a group.

$\blacksquare$


Also see

From Subgroup is Normal iff Left Cosets are Right Cosets, the left coset space of a normal subgroup is equal to its right coset space.


It follows that $G / N$ does not depend on whether left cosets are used to define it or right cosets.

Thus we do not need to distinguish between the left quotient group and the right quotient group - the two are one and the same.


Sources

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