Quotient Group

For a group G and a normal subgroup N of G, the quotient group of N in G, written G/N and read "G modulo N", is the set of cosets of N in G. Quotient groups are also called factor groups. The elements of G/N are written Na and form a group under the normal operation on the group N on the coefficient a. Thus,

Since all elements of G will appear in exactly one coset of the normal subgroup N, it follows that

where |G| denotes the order of a group. This is also a consequence of Lagrange's group theorem with H=G and K=N

Although the slash notation conflicts with that for an extension field, the meaning can be determined based on context.

See also

Abhyankar's Conjecture, Backslash, Coset, Extension Field, Lagrange's Group Theorem, Outer Automorphism Group, Normal Subgroup, Subgroup

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References

Herstein, I. N. Topics in Algebra, 2nd ed. New York:Springer-Verlag, 1975.

Referenced on Wolfram|Alpha

Quotient Group

Cite this as:

Weisstein, Eric W. "Quotient Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuotientGroup.html

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