Definition:Group

A group is a nonempty set (which we will call $$G$$) coupled with a binary operation (which we will denote as $$\cdot$$) which satisfies the following four properties:


 * Closure: For each $$a,b \in G,$$ $$a \cdot b$$ is an element of $$G$$.
 * Associativity: For all $$a,b,c \in G,$$ $$(a \cdot b) \cdot c = a \cdot (b \cdot c)$$.
 * Identity: There exists an element $$e \in G$$ such that $$a \cdot e=e \cdot a=a$$ for each $$a \in G$$.  This element is called the identity.
 * Inverses: For each $$a \in G,$$ there exists an element $$b \in G$$ such that $$a \cdot b=b \cdot a=e$$.  This element is called the inverse of $$a$$ and is denoted $$a^{-1}$$.

The notation $$(G,\cdot)$$ is used to represent a group.

Trivial Group
The trivial group is a group with only one element $$e$$.

For a group $$G = \left\{{e}\right\}$$ to be a group, it follows that $$e \cdot e = e$$.

Showing that $$\left({G, \cdot}\right)$$ is in fact a group is straightforward:


 * $$G$$ is closed
 * $$e$$ is the identity
 * $$e \cdot \left({e \cdot e}\right) = e = \left({e \cdot e}\right) \cdot e$$ so $$\cdot$$ is associative
 * $$e \cdot e = e \Longrightarrow e^{-1} = e$$ and thus every element of $$G$$ (all one of them) has an inverse.