In response to a question from Ghodratollah Aalipour in Tehran, I have been thinking again about the power graph of a group. This is the graph whose vertices are the group elements, two vertices being joined if one is a power of the other.
I can prove:
Proposition. The clique number of the power graph of any group is at most countably infinite.
The proof is not difficult. Let C be a clique containing the element x. Then we can divide the rest of C into two parts: elements y such that y=xn; and elements y such that x=yn. The first part is obviously countable. For the second, we show that the set of such elements with any fixed value of n is finite, from which the result will follow. If x=yn=zn and y and z are joined (so one is a power of the other), a short calculation shows that y and z generate the same, finite, cyclic group, and we are done.
Aalipour and his colleagues had proved that if G has exponent n, then its clique number is at most n, and had asked me whether the same is true for the chromatic number. (Indeed, it is.) But this suggests an open problem:
Problem. Is it true that the chromatic number of the power graph of any group is at most countably infinite?