A quotient group of by , where is a group and is a normal subgroup of , is the set of cosets of in . It is read " modulo ". Quotient groups are also called factor groups. The elements of are written and form a group under the operation induced by the group operation in . Thus,

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

where denotes the group order, interpreted as a cardinal number for an infinite group. This is also a consequence of Lagrange's group theorem with and , where is the identity element of .

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, Cokernel, Coset, Extension Field, Lagrange's Group Theorem, Normal Subgroup, Outer Automorphism Group, 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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