Group

Last updated Nov 1, 2022

Suppose $X$ is a Nonempty Set and $+: X \times X \to X$ is an Operation on $X$. Then $X$ is a Group if it satisfies the following properties:

1. Additive Identity: There exists $0 \in X$ so that for all $x \in X$, $0 + x = x$.
2. Associativity of $+$: For all $x,y,z \in X$, $(x + y) + z = x + (y + z)$
3. Additive Inverses: If $x \in X$, then there exists $-x \in X$ so that $x + (-x) = 0 = (-x) + x$

1. In other words, a Group is a Monoid with Additive Inverses.
2. When the context is clear, we often write $a + b$ as $ab$. This helps clear up confusion with the usual understanding of $+$.
3. If a Group also satisfies Commutativity of $+$, then it is an Abelian Group.
4. The inverse in a Group is unique