a **group** is a [[monoid]] with *inverses* $\forall u : \exists v : u \cdot v = v \cdot u = e$ ![[monoid]] Given a set and an operation: must check: - closure - associativity - identity - inverses If the operation is commutative we call the group [[abelian group|abelian]]. A [[group action]] describes how a group can be thought of as *operating* on some space. > [!NOTE]- inverses are unique > > Let $v, w$ be inverses of $u$. Then $uv = uw = e$. Then $vuv = vuw$. So $v = w$. > [!NOTE]- left and right inverses coincide > Let $vu = e$. We want to show $uv = e$. > > We know $u$ has some inverse $w$. We can use this inverse to show that > > $\begin{align*} > (uv)u &= u \\ > (uv)(uw) &= e \\ > uv &= e > \end{align*}$ > > Combined with the above this implies that $u$ has a unique inverse $u^{-1}$. > [!NOTE]- double inverse is identity > > Let $v = u^{-1}$. We want to show $v^{-1} = u$. This is trivial since $uv = e$. a group can be thought of as a category with a single object. each element of the group corresponds to an [[isomorphism]] and the identity element obviously corresponds to the identity morphism. the group operation is given by arrow composition. > [!NOTE]- inverse of product is (reversed) product of inverses > > $\begin{align*} > uv(uv)^{-1} &= e \\ > v (uv)^{-1} &= u^{-1} \\ > (uv)^{-1} &= v^{-1} u^{-1} > \end{align*}$ > > Same proof as [[composition of inverses]] under the above perspective. # examples $(\mathbb{Z}, +)$ $(\mathbb{R}, +)$ [[vector space]]s under addition (by definition) $(\mathbb{R}_{> 0}, \times)$ $(\mathbb{C} \backslash \{ 0 \}, \times)$ $(\mathbb{S}^{1}, \times)$ ([[sphere|complex unit circle]]) $(\mathbb{Q} \backslash \{ 0 \}, \cdot)$ [[invertible linear map|general linear group]] [[automorphism|symmetric group]] # sources [Groups — A Primer – Math ∩ Programming](https://jeremykun.com/2012/12/08/groups-a-primer/) https://mathworld.wolfram.com/AdditiveGroup.html