Conjugate elements

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


 * 1) There is an inner automorphism of the group mapping one element to the other
 * 2) 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:


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

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