Group theory and generalizations Definition

Quotient group

Builds on Normal subgroup Left coset Normality via left and right cosets

Definition

Let $G$ be a group and let $N$ be a normal subgroup of $G$ . The quotient group $G/N$ is the set of left cosets $\{gN : g \in G\}$ with the operation $(aN)(bN) = (ab)N$ .

Remarks

The operation is well defined because $N$ is normal: if $aN = a'N$ and $bN = b'N$ , then $abN = a'b'N$ . The identity of $G/N$ is $N = eN$ , and the inverse of $aN$ is $a^{-1}N$ . When $G$ is finite, $|G/N| = [G : N] = |G| / |N|$ . The map $g \mapsto gN$ is a surjective homomorphism from $G$ to $G/N$ with kernel $N$ .

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Group concept
Conjugate elements definition
Subgroup definition
Conjugate subgroup definition
Left coset definition
Right coset definition
Normal subgroup definition
Normality via left and right cosets statement
Quotient group definition

Where this leads

59 pages build on this one.

Tag	`0YYWCNTPWR`
URI	`m2://0YYWCNTPWR`
Kind	definition
Writing	W2 usable
MSC	20A05
Defines	Quotient group
Uses	Normal subgroup, Left coset, Normality via left and right cosets

Quotient group

Definition

Remarks

Reading path

Where this leads

Further reading

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