Group theory and generalizations Proposition

Normality via left and right cosets

Builds on Normal subgroup Left coset Right coset

Statement

Let $G$ be a group and let $N$ be a subgroup of $G$ . Then $N$ is normal in $G$ if and only if $gN = Ng$ for each $g \in G$ .

Remarks

Here $gN$ is a left coset and $Ng$ is a right coset. The equation $gN = Ng$ is equivalent to $gNg^{-1} = N$ , by multiplying on the right by $g^{-1}$ . The condition does not say that $gn = ng$ for each $n \in N$ ; it says only that the two sets are equal.

Reading path

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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

Where this leads

62 pages build on this one.

Tag	`01E7SWYJX3`
URI	`m2://01E7SWYJX3`
Kind	statement, proposition
Writing	W0 stub
MSC	20A05
Status	This copy of the site has no build data. Run `m2 status 01E7SWYJX3`.
Uses	Normal subgroup, Left coset, Right coset

Normality via left and right cosets

Statement

Remarks

Reading path

Where this leads

Further reading

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