Conjugation
المؤلف:
Fraleigh, J. B
المصدر:
A First Course in Abstract Algebra, 7th ed. Reading, MA: Addison-Wesley, 2002.
الجزء والصفحة:
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27-11-2018
603
Conjugation
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.
REFERENCES:
Fraleigh, J. B. A First Course in Abstract Algebra, 7th ed. Reading, MA: Addison-Wesley, 2002.
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