Group Theory

Let $G \neq \empty$ be a set and $\circ$ be a binary operation on $G$.

$(G, \circ)$ is called a group if it satisfies the following.

1. For all $a, b \in G$, $a \circ b \in G$ (closure)
2. For all $a, b, c, \in G$, $a \circ (b \circ c) = (a \circ b) \circ c$ (associativity)
3. There exists $e \in G$ with $a \circ e = e \circ a = a$ for all $a \in G$ (identity or unit element)
4. For each $a \in G$, there is an element $b \in G$ such that $a \circ b = b \circ a = e$ (inverse)

$G$ is commutative or abelian if $a \circ b = b \circ a$ for all $a, b \in G$

• is right inverse different from left inverse?

Examples of Groups

Under ordinary $+$, $(\mathbb{Z}, +)$, $(\mathbb{Q}, +)$, $(\mathbb{R}, +)$, $(\mathbb{C}, +)$ are groups

• inverse of $a$ is simply $-a$, which exists

Under ordinary $+$, $(\mathbb{N}, +)$ is not a group.

• inverse of $a \in \mathbb{Z}^+$ does not exist

Under ordinary $\times$, none of $(\mathbb{Z}, \times)$, $(\mathbb{Q}, \times)$, $(\mathbb{R}, \times)$, $(\mathbb{C}, \times)$ are groups

• number of 0 has no inverses.

Under ordinary $\times$, $(\mathbb{Q}^, \times)$, $(\mathbb{R}^, \times)$, $(\mathbb{C}^*, \times)$ are groups.

• $A^*$ denotes the nonzero elements of $A$

Under ordinary $-$, $(\mathbb{Z}, -)$, $(\mathbb{Q}, -)$, $(\mathbb{R}, -)$ are not groups

• associative axiom fails → $a - (b - c) \neq (a - b) - c$