Let $P(E)$ denote the power set of a set $E$: the set of subsets of $E$. Does the operation $A\cap B$ define a structure of group?
By denition, a group $G$ is a set with an operation $g.h$ (formally, a function $G\times G\rightarrow G$), with the following properties:
The property of the identity: for all $g\in G$, $e.g = g.e = g$.
Existence of inverses: for all $g\in G$ there is $h\in G$ (the inverse of $g$) such that $h.g = g.h = e$.
Associativity: for all $x,y,z\in G$, $x.(y.z) = (x.y).z$.
If the operation $g.h$ is commutative, that is, if $g.h = h.g$ for all $h,g\in G$ then the group is said to be abelian.
Τhanks in advanced!