Intuitive Explanation of Group Conjugation So I've studied abstract algebra and mainly focused on finite groups and touched on rings and fields for a bit which was cool. Anyways, most of the theory made sense except for the congugate of an element of a group. I don't know what it is, but I find the definition really confusing and was hoping someone could help me a little.
By definition, two elements $a$ and $b$ of a group $G$ are the conjugate of one another if there exists $g\in G$ such that $gag^{-1}=b$.
I can see how they form an equivalence class by using the identity and composition of elements, but beyond that it seems bizarre. What is the definition really getting at? Where would one use the definition—perhaps in the construction of a quotient group?
Warm regards,
Alex
 A: Conjugation by a group element provides an example of a group automorphism.  What this means is that if elements $g$ and $h$ are conjugate, they essentially act the same in the group.  Take for example the symmetric group $S_{3}$.  There are three elements of order $2$: $(1,2)$, $(1,3)$, and $(2,3)$.  They are all conjugate.  They all essentially behave the same, in fact by relabelling the symbols we are permuting we see that these three group elements are in some ways indistinguishable.
In addition to talking about group elements being conjugate, we are also interested in when subgroups are conjugate.  This means that you essentially take the whole collection of elements of the subgroup and conjugate them all to obtain a different subgroup.  If we are interested in describing all of the subgroups, we would usually just give one representative from each conjugacy class, that gives us really all of the information we need. 
Likewise with group elements, if we are looking at permutations of order 2 acting on a set with let's say 5 elements, we could focus on taking one group element of order 2 from each conjugacy class to describe all of the possible group actions we are interested in. These would have the form $(1,2)$, or else $(1,2)(3,4)$; this describes all of the possibilities up to conjugacy. 
A: If you think of a group as a set of symmetries of some object then two elements are conjugate when they are symmetries of the same type. For example, the six element symmetric group $S_3$ is (essentially) the set of symmetries of an equilateral triangle. The three reflections form a conjugacy class. So do the two nontrivial rotations. The identity is in a class by itself (that's always true).
Cayley's theorem says that any group can be represented as a set of symmetries, so this interpretation isn't as limited as it might seem. (See https://en.wikipedia.org/wiki/Cayley's_theorem .)
