I have the following notations that I don't understand what it stands for? A is a group 
what is that weird looking $p$ does it mean the all of the subsets of A in pairs like $(a_1,a_2)$ while $a_1\in A$ and $a_2 \in A$ what is the different of that and 
this $p$ is different does it mean all of the subset of group $\mathbb N$
I'd be really glad if someone could clear this, I'm confused.
Any help will be appreciated, thanks.
The symbol
$\mathcal P$
stands for the power set, i.e. $$\mathcal P(X)$$ is the set of all subsets of $X$ (also sometimes written as $2^X$).
The symbol $\times$ denotes the cartesian product, i.e. $$X\times Y = \{(x,y)| x\in X, y\in Y\}$$ is the set of all pairs of elements from $x$ and $y$.
So, $$\mathcal P(A\times A)$$ is the set of all subsets of pairs of elements of $A$. For example, if $A=\{1,2\}$, then $A\times A = \{(1,1),(1,2),(2,1),(2,2)\}$ and $$\mathcal P(A\times A) = \\ \{\{\}, \{(1,1)\},\{(1,2)\},\{(2,1)\},\{(2,2)\},\\ \{(1,1),(1,2)\}, \{(1,1),(2,1)\}, \{(1,1),(2,2)\}, \{(1,2),(2,1)\}, \{(1,2),(2,2)\}, \{(2,1),(2,2)\}, \\ \{(1,1),(1,2),(2,1)\}, \{(1,1),(1,2),(2,2)\}, \{(1,1),(2,1),(2,2)\}, \{(1,2),(2,1), (2,2)\},\\ \{(1,1),(1,2),(2,1),(2,2)\} \}$$