Groups

Definition: Group

A group is a monoid $(G,\circ )$ which also satisfies the existence of inverse elements:

• For each $a\in G$, there exists some $a_{\text{inv}}\in G$ such that $a\circ a_{\text{inv}}=a_{\text{inv}}\circ a=e$, where $e$ is the identity element of the monoid.

Theorem: Uniqueness of Group Inverses

Each element of a group has a unique inverse.

PROOF

Let $a$ be an element of a group $(G,\circ )$.

Suppose there are two distinct inverses $a_{\text{inv}}$ and $a_{\text{inv}}^{\prime }$ of $a$. By definition,

$$a\circ a_{\text{inv}}=e$$ $$a\circ a_{\text{inv}}^{\prime }=e$$

Therefore,

$$a\circ a_{\text{inv}}=a\circ a_{\text{inv}}^{\prime }$$

We multiply both sides by $a_{\text{inv}}$ on the left and use associativity.

$$a_{\text{inv}}\circ a\circ a_{\text{inv}}=a_{\text{inv}}\circ a\circ a_{\text{inv}}^{\prime }$$ $$(a_{\text{inv}}\circ a)\circ a_{\text{inv}}=(a_{\text{inv}}\circ a)\circ a_{\text{inv}}^{\prime }$$ $$e\circ a_{\text{inv}}=e\circ a_{\text{inv}}^{\prime }$$ $$a_{\text{inv}}=a_{\text{inv}}^{\prime }$$

NOTATION

There are two notations used for groups and both are equivalent.

Additive Notation

The group operation is denoted by $+$, the identity element by $0$ and the inverse of each $g\in G$ by $-g$. Instead of writing $g+-g$ or $g+(-g)$, we write simply $g-g$.

Multiplicative Notation

The group operation is denoted by $\cdot$, the identity element by $1$ and the inverse of each $g\in G$ by $g^{-1}$. We can also write simply $ab$ instead of $a\cdot b$.