Can a group be defined in terms of a relation on a set? Wikipedia defines a group as "an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element." I keep thinking that there is a connection to this definition and a relation on a set, but I'm not sure what it is. Obviously, relations and operators are connected. Can groups be defined in terms of sets and relations? I am new to this, and the Wikipedia article is over my head.

A group can be defined as a set and a relations. 
Note that a relation $R$ on a set $G$ is any subset of $G \times G$. A function is a relation on $G \times G$ such that if $(a,b) \in f$ and $(a,c) \in f$, then $b = c$.
Therefore a group is a set $G$ with a relation $*$ which happens to be a function. Moreover, this function satisfies some properties like associativity, etc. Also as is typical in model theory, you often say a group is a set, with a binary function $*$, and a constant $e$, which represents the identity. Again, the constant can still be thought of as a unary relation. You can also define a group to include a symbol for taking an inverse. This can still be thought of as a relation since it is a unary function. 
