Conjugation is the process of taking a complex conjugate of a complex number, complex matrix, etc., or of performing a conjugation move on a knot.

Conjugation also has a meaning in group theory. Let be a group and let . Then, defines a homomorphism given by

This is a homomorphism because

The operation on given by is called conjugation by .

Conjugation is an important construction in group theory. Conjugation defines a group action of a group on itself and this often yields useful information about the group. For example, this technique is how the Sylow theorems are proven. More importantly, a normal subgroup of a group is a subgroup which is invariant under conjugation by any element. Normal groups are extremely important because they are the kernels of homomorphisms and it is possible to take the quotient of a group and one of its normal subgroups.

See also

Complex Conjugate, Complex Matrix, Complex Number, Conjugacy Class, Conjugate Element, Conjugate Matrix, Conjugate Subgroup, Conjugation Move, Normal Subgroup, Similar Matrices, Similarity Transformation, Sylow Theorems

Portions of this entry contributed by John Renze

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References

Fraleigh, J. B. A First Course in Abstract Algebra, 7th ed. Reading, MA: Addison-Wesley, 2002.

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Conjugation

Cite this as:

Weisstein, Eric W., with contributions by John Renze. "Conjugation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Conjugation.html

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