Conjugate elements: Difference between revisions

Latest revision as of 22:43, 9 June 2008

This article describes an equivalence relation on the set of elements of a group

Definition

Symbol-free definition

Given a group, two (possibly equal) elements of the group are termed conjugate elements if the following equivalent conditions are satisfied:

There is an inner automorphism of the group mapping one element to the other
There are two elements of the group whose products, in the two possible orders, give these two elements

Definition with symbols

Given a group $G$ and elements $g, h \in G$ , $g$ is termed conjugate to $h$ if the following equivalent conditions are satisfied:

There exists $x \in G$ such that $x g x^{- 1} = h$ , in other words, the inner automorphism of conjugation by $x$ , sends $g$ to $h$
There exist $a, b \in G$ such that $g = a b, h = b a$

The equivalence classes under the equivalence relation of being conjugate are termed the conjugacy classes.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of conjugate elements

@@ Line 5: / Line 5: @@
 ===Symbol-free definition===
-Given a group, two (possibly equal) elements of the group are termed '''conjugate elements''' if there is an [[inner automorphism]] of the group mapping one element to the other.
+Given a group, two (possibly equal) elements of the group are termed '''conjugate elements''' if the following equivalent conditions are satisfied:
+# There is an [[inner automorphism]] of the group mapping one element to the other
+# There are two elements of the group whose products, in the two possible orders, give these two elements
 ===Definition with symbols===
-Given a [[group]] <math>G</math> and elements <math>g,h \in G</math>, <math>g</math> is termed conjugate to <math>h</math> if there exists <math>x \in G</math> such that <math>xgx^{-1} = h</math>, in other words, the [[inner automorphism]] of conjugation by <math>x</math>, sends <math>g</math> to <math>h</math>.
+Given a [[group]] <math>G</math> and elements <math>g,h \in G</math>, <math>g</math> is termed conjugate to <math>h</math> if the following equivalent conditions are satisfied:
+# There exists <math>x \in G</math> such that <math>xgx^{-1} = h</math>, in other words, the [[inner automorphism]] of conjugation by <math>x</math>, sends <math>g</math> to <math>h</math>
+# There exist <math>a,b \in G</math> such that <math>g = ab, h = ba</math>
 The equivalence classes under the equivalence relation of being conjugate are termed the [[conjugacy class]]es.
+===Equivalence of definitions===
+{{proofat|[[Equivalence of definitions of conjugate elements]]}}