There is one way to understand conjugacy from the perspective of linear algebra, using change-of-basis. Conjugacy can also be interpreted as a "measure of departure" from being a commutative group.
4 There are several reasons in finite group theory. Conjugacy is a equivalence relation, and therefore breaks up the group as a disjoint union of equivalence classes called conjugacy classes.
Conjugacy classes are the orbits of elements of the group, under the action of conjugation. It has nothing to see with subgroups. For instance, in an abelian group, the conjugacy classes are simply the singletons made up of the elements of the group, while a subgroup usually has more than one element…
4 One way to think about this problem is the following: think of conjugacy classes as group elements up to change of basis. The identity transformation is in a single conjugacy class. Any reflection about a diagonal is in a single conjugacy class. Any reflection without fixed points (i.e. a reflection through the middle of opposite edges) is ...
2 Conjugacy is one of the more important concepts in group theory. It's an equivalence relation, an easy exercise. The class equation gives, for a finite group, the sizes of the various conjugacy classes. This equation is very useful.
First, you can find the size of the conjugacy class of $ (1234567)$. Then, you have probably seen a theorem relating the size of the conjugacy class to the size of the centralizer; use this to find the size of the centralizer. Now, there are some elements that "obviously" commute with $ (1234567)$; to begin with, its powers, and any permutation disjoint from it. Compare the number of these ...
Here is the "conjugation table" of $Q$, where each element in the table is equal to the row header conjugated by the column header, and the end of each row is the conjugacy class of the row header.
We get the conjugacy equation (and of course the identity element has orbit only itself) $2 n=1 + \underbrace {2 +2+ \cdots +2} _ {m \rm {\,times}} +n.$ For the even case the analysis is the same but we get some different type of orbits, which is expected since the even dihedral has center whereas the odd does not.
In group theory, the mathematical definition for "conjugation" is: $$ (g, h) \\mapsto g h g^{-1} $$ But what exactly does this mean, like in laymans terms?
So, given the order, the group with least number of conjugacy classes are the abelian groups of that order. Now that you know that conjugation is an equivalence relation, and that equivalence relation partitions the set into disjoint sets, you now have a new way looking at the order of the group.
