On the domination number of proper power graphs of finite groups

The proper power graph $P^{⁎}\left(G\right)$ of a finite group G is the simple graph whose vertices are the nonindentity elements of G and two distinct vertices are adjacent if one of them is a power of the other. In this paper, we study the domination number $\gamma \left(P^{⁎}\right(G\left)\right)$ of $P^{⁎}\left(G\right)$ by relating it with the number of distinct prime order subgroups of G. For a nilpotent group G, we give a sharp upper bound for $\gamma \left(P^{⁎}\right(G\left)\right)$. When G is a direct product of two nontrivial groups H and K, we give a sharp lower bound for $\gamma \left(P^{⁎}\right(G\left)\right)$ in terms of the number of components of $P^{⁎}\left(H\right)$ and $P^{⁎}\left(K\right)$. As an application, we determine $\gamma \left(P^{⁎}\right(G\left)\right)$ when G is a nilpotent group whose order is divisible by at most two distinct primes.