# Algebra of Complexes of a Group

Let us consider the set of all complexes of a group $G$, which in nothing but power set of $G$. Let it be denoted by $P\left( G \right)$. Now, we define three binary composition in $P\left( G \right)$. The two compositions namely union and intersection of sets are familiar ones. How we define the multiplication of complexes.

Multiplication is Complexes
Let $H$ and $K$ be two complexes of a group $G$ whose composition has been denoted multiplicatively, then the product of $H$ and $K$ denoted by $HK$ is defined as

In other words $HK$ is the set of all possible products of elements of $H$ with those of $K$. It is evident that $hk \in HK$.

Thus the product of two complexes is also a complex of the group.

Multiplication of Complexes is Associative
Let $H,K$ and $L$ be three complexes of a group $G$ whose composition is denoted multiplicatively. Then

Because $\left( {hk} \right)l = h\left( {kl} \right)$, multiplication in $G$ being associative

Also

Inverse of Complexes in a Group
Let $H$ be any complex of $G$ and let us define

Then ${H^{ - 1}}$ is the complex of $G$ consisting of the inverse of the elements of $H$. This ${H^{ - 1}}$ is called the inverse of complex $H$.