Implications of Lagrange's theorem

In a general semigroup, the order of any subsemigroup doesn't need to divide the order of the semigroup. As a consequence, there are little to no limits placed on the types of graphs that can be commuting graphs of semigroups. On the other hand, by Lagrange's theorem the order of any subgroup of a group divides the order of the group. This implies that the size of any commutatively necessary subsemigroup must divide the order of the group. Therefore, from Lagrange's theorem alone we can get these restrictions placed on the class of commuting graphs of groups:

• The degree of any vertex must divide the order of the graph
• The total number of dominant vertices must divide the order of the graph
• The clique number must divide the order of the graph
• In general, any commutatively-necessary subalgebra must divide the order of the graph

It is clear from this that there is a significant difference between the commuting graphs of groups and semigroups. All sorts of semigroups with commuting graphs unlike anything that could emerge from a group exist (like rectangular bands for example). The starting point of the theory of commuting graphs of groups should be the implications of Lagrange's theorem on the possible commuting graphs of a group, but there is more that can be infered beyond that.

Beyond Lagrange's theorem:
Groups are strongly divisibility commutative, and this is characterized by the conjugacy classes of elements, which are always non-empty. The first thing we can say beyond Lagrange's theorem is that the size of the conjugacy classes of the group are equal to the index of the centralizer. The size of each of these conjugacy classes must then sum up to equal the order of the group. We can from this, for example that groups of prime-squared order are necessarily commutative.