Please Login to access more options.

Problem (The Class Equation)

Prove the following corollaries of The number of conjugates of $a$

  1. For any group $G$, we have $$|G|=\sum|G:C(a)|$$ where the sum runs over one element from each conjugacy class of $G$.
  2. The center of any group consists precisely of the elements $a\in G$ such that $\text{cl}(a)=\{a\}$. In symbols, we have $a\in Z(G)$ if and only if $|\text{cl}(a)|=1$.
  3. For any group $G$, we know that $$|G|=|Z(G)|+\sum|G:C(a)|$$ where the sum runs over one element from each conjugacy class of $G$ that has more than one element.
  4. If $G$ has order $p^k$ for some prime $p$ and $k\in \mathbb{N}$, then $p$ divides $|Z(G)|$, and so the center is nontrivial.

The following pages link to this page.