A group is a form of [[Algebraic Structure|Algebraic System]] that is defined as a [[Empty Set|nonempty]] [[Set|set]] $G$ equipped with a [[Binary Operation|binary operation]] $\circ$ such that that the following [[Axiom|axioms]] hold (given some $x,y,z \in G$): - $\circ$ is [[Associative Property|Associative]], $(x \circ y) \circ z = x \circ (y \circ z)$ - $\circ$ is [[Closure|closed]] under $G$, $\nexists x,y\in G: x \circ y \notin G$ - There exists an [[Identity Function|identity]] element $e \in G$, $x \circ e = e \circ x = x$ - Each element $a\in G$ has an [[Inverse Function|inverse]] element in $G$ denoted $a^{-1}$ such that composing them equals the identity, $a \circ a^{-1} = e$ >[!note] >It can be proven downstream from these axioms the following: >$\huge \begin{align} > a &= x >\end{align} >$ A group can be denoted as $[G; \circ]$, but is also denoted simply as its [[Underlying Set|underlying set]] $G$ for brevity if the operation is understood. ### Groups as Symmetries Numbers are an abstraction of literaly counts of objects, and in the same vein a way of thinking groups is relating them as an abstraction over symmetry. Here is an example of the [[Dihedral Group|dihedral group]] $D_4$, which is the group of symetries of a square. ![[Screenshot 2025-10-20 at 11.15.18 PM.png|700]] This collection, as mentioned before, is a [[Dihedral Group|group]] $D_4$ whose [[Set|set]] is a collection of [[Symmetric|symmetric]] actions on a [[Square|square]] that leave it identical to how it was. The operation for $D_4$ is [[Composition|composition]]. An example of composing two elements in this group to get another would be rotating by $90\degree$ followed by a $180\degree$ rotation to get a state identical to just rotating $270\degree$. $\huge r_{1} \circ r_{2} = r_{3}$ In this group, the identity element would be $r_0$, which represents rotating by $0\degree$, which is the action of doing nothing. $ \huge r_{1} \circ r_{3} = r_{0} $