Enhanced power graphs

The enhanced power graph is a subgraph of the commuting graph. Two elements are adjacent in the enhanced power graph if they are in the same monogenic subsemigroup. As monogenic semigroups are totally commutative, all pairs commute. This leads to following partially ordered family of graphs:

Both the commuting graph and the enhanced power graph can be defined subalgebraically. In general, if we have any subalgebraically closed class $C$ of algebras, then for any algebra $A$ we can get a lower set of the lattice $Sub(A)$ consisting of all subalgebras in that class. A graph can then be constructed from the primal graph of that lower set. This graph-theoretic technique of universal algebra opens up an infinite set of possible graphs that can be defined for any algebra.

The enhanced power graph is always a subgraph of the commuting graph, however, in certain conditions they can coincide. Consider that in order for them to commutative subsemigroups need to be monogenic. Restricting to the case of groups, there is a characterization of groups for which every commutative subgroup is cyclic.

Finite groups with periodic cohomology:
Groups in which every abelian subgroup is cyclic are considered to have periodic cohomology. In these groups, the Sylow p-subgroups are all either cyclic or generalized quaternion groups. The quaternion group is therefore the simplest case of a p-group for which the enhanced power graph coincides with the commuting graph.

Links:
Finite groups with periodic cohomology