On the power graphs of elementary abelian and extra special $p$-groups‎

‎For a given odd prime $p$‎, ‎we investigate the power graphs of three classes of finite groups‎: ‎the elementary abelian groups of exponent $p$‎, ‎and the extra special groups of exponents $p$ or $p^۲$‎. ‎We show that these power graphs are Eulerian for every $p$‎. ‎As a corollary‎, ‎we describe two classes of non-isomorphic groups with isomorphic power graphs‎. ‎In addition‎, ‎we prove that the clique graphs of the power graphs of two considered classes are complete‎.

Keywords:

‎Power graphs‎ , ‎non-group semigroups‎ , ‎semidirect product