Q8's conjugacy classes are {e}, {a, a^3}, {a^2, a^2b}, and {b, ab, a^3b}. Centralizers include Q8, cyclic subgroups, and the Klein four-group. Common centralizers indicate shared conjugacy classes.
The quaternion group Q8 consists of the elements {e, a, a^2, a^3, b, ab, a^2b, a^3b}. To find conjugacy classes, we observe how elements transform under conjugation by other group elements.
Conjugacy Classes:
Conjugate elements form a class. In Q8, conjugacy classes are {e}, {a, a^3}, {a^2, a^2b}, and {b, ab, a^3b}, based on powers of a and their products with b.
Centralizers:
The centralizer of an element x is the set of elements that commute with x: C_G(x) = {g in G : gx = xg}, where G is the group.
In Q8, centralizers are:
For e, the entire Q8.
For a, a^3, b, and a^3b, their centralizers are cyclic subgroups they generate (⟨a⟩, ⟨b⟩, ⟨a^2⟩, ⟨ab⟩).
For a^2 and a^2b, their centralizers are the Klein four-group {e, a^2, b, ab}.
Finding centralizers is crucial for determining conjugacy classes. Elements in the same conjugacy class share the same centralizer, revealing common algebraic properties. The centralizer structure aids in identifying conjugacy classes.
The question probable may be:
Let Q8={e,a,a2,a3,b,ab,a2b,a3b}. Find all conjugacy classes of Q8 and the corresponidy centralizers. What does finding the centralizers have to do with finding the conjugacy classes