Fix a prime ${p}$. We say that two finite groups $G$ and

$H$ are ``${p}$-equivalent'' if there is an isomorphism between

Sylow $p$-subgroups $S \in Syl_p(G)$ and $T\in

Syl_p({H})$ that preserves all $G-$ and

${H}-$conjugacy relations among elements and subgroups of $S$

and $T$. We say that two topological spaces ${X}$ and ${Y}$ are ``${p}$-equivalent'' if there is a third space ${Z}$, and maps $X\to Z \leftarrow Y$ that induce isomorphisms in homology

with coefficients in $\mathbb{Z}/p$. (Both of these are equivalence

relations.) The main theorem I want to describe says that finite groups ${G}$ and ${H}$ are

$p$-equivalent (as groups) if and only if their classifying spaces

are ${p}$-equivalent (as spaces).

I will start by defining in more detail classifying spaces of discrete

groups and the two kinds of ${p}$-equivalence described above, and

also saying a little about the background of the theorem. I then plan to

give some examples of finite groups that are ${p}$-locally equivalent

but not isomorphic, and say something about ideas that went into the

proof of the theorem (carried out by several different people over a

period of 10--15 years).