TITLE: The use of conjugacy class and centralizer?
QUESTION [15 upvotes]: This is more or less for a conceptual and better-understanding question in group theory and in representation theory:

(1) Why are conjugacy class and centralizer important concepts in the group / representation theory? What is the important use of conjugacy class and centralizer of an element $g$ in a group $G$?

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(2) How is the case of the conjugacy class and centralizer for an element $g$ in a finite group $G$, different from a continuous group $G'$?

REPLY [2 votes]: One of the most important concepts in representation theory is induction and restriction. The Frobenius character formula is a useful tool to understand this and has many generalizations to continuous groups (Poisson summation formula, Selberg trace formula, Arthur trace formula). It gives you an identity between irreducible representations with multiplicities and conjugacy classes with volumes of centralizers.