# Definition A collection of elements $G$ together with a binary operation $\bullet$ is called a *group* if it satisfies the axioms: 1. *Associativity*: For $g_1, g_2, g_3 \in G, \quad g_1 \bullet (g_2 \bullet g_3) = (g_1 \bullet g_2) \bullet g_3$ 2. *Right identity:* $G$ contains an element $e$ such that, for any $g$ in $G$, $g \bullet e = g$ 3. *Right inverse:* for every $g$ in $G$ there is an element called $g^{-1}$, also in $G$, for which $g \bullet g^{-1} = e$ 4. *Closure*: for $g_1, g_2 \in G$, $g_1 \bullet g_2 \in G$ Note that from (1-3) we can conclude that $e$ is unique, and that it is also a left-identity (i.e. $e \bullet g = g$). The inverse element $g^{-1}$ is also unique for any $g$, and it is also a left inverse (i.e. $g^{-1} \bullet g = e$). If all [[Group Element|elements]] of the group commute, i.e. $g \bullet h = h \bullet g, \quad \forall g, h \in G$, then it is an [[Abelian Group|abelian (or commutative) group]]. Otherwise, it is said to be non-abelian or non-commutative. Intuitively speaking, a group is just a set of transformations which are *composable* and *invertible*. Usually, to prove that a set equipped with a binary operation is a group, we only need to prove closure and that it contains all its inverses. The remaining two axioms are often trivial. We will often omit the $\bullet$ and use the notation $gh \equiv g \bullet h$ when it is clear from the context. # Cancellation Laws It is straightforward to prove the following cancellation laws $\forall g_1, g_2, h \in G$ $ g_1 \bullet h = g_2 \bullet h \implies g_1 = g_2 h \bullet g_1 = h \bullet g_2 \implies g_1 = g_2 $ # In Physics In physics, we often think of a group as a set of transformations. In that case, the group's binary operation is just composition.