class number of groups and rings

Question

The class number of a group is defined by the number of conjugacy classes.

The class number of a ring(specifically a number field or its algebraic integers) is defined by the order of ideal class group.

Is there any relation between them?

Answer 1

The ideal class group of a number field is abelian, hence its conjugacy classes are singletons and the two notions of "class number" you defined coincide.

In my opinion a more interesting connection between conjugacy classes and ideal class groups is given by a theorem of Latimer and MacDuffee:

${\bf Theorem}$: Let $f$ be a monic irreducible polynomial with integer coefficients which has degree $n$. Then conjugacy classes of matrices in $\mathrm{M}_n(\mathbb Z)$ with characteristic polynomial $f$ are in bijection with the $\mathbb Z[\alpha]$-ideal classes in $\mathbb Q(\alpha)$, where $\alpha$ is a root of $f$.

Keith Conrad has a nice blurb about this result on his website.