Group
Undergraduate (abstract algebra)
Consider a set $G$ with a binary operation $\ast$. The set $G$ is a group if
- $\ast$ is associative: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in G$,
- $G$ has an identity element $e$ such that $e \ast g = g \ast e = g$ for all $g \in G$, and
- every element $g \in G$ has an inverse $g^{-1}$ such that $g \ast g^{-1} = g^{-1} \ast g = e$.
Examples
Examples
-
$\mathbb{Z}$ with addition
High school
Adding $0$ changes nothing, and adding $-3$ undoes adding $3$.
Non-examples
Things that may be mistaken for examples.
-
$\nats$ with addition
High school
$\nats$ with addition is not a group, because no positive integer has an additive inverse in $\nats$.
Properties
- The identity element of a group is unique, and so is the inverse of each element.
Notes
In mathematics, “group” is a technical term. Avoid using it informally to mean “a bunch of objects.”
Distinctions involving this term
- Types of collections: Mathematics has several kinds of “a bunch of things,” which differ in whether order and repetition matter. A set ignores order and repetition. A multiset keeps repetition. A sequence keeps order and repetition. “Collection” is informal, and “group” is a technical term for a set with an operation.