---- > [!definition] Definition. ([[generating set of a group]]) > A **generating set of a group** is a subset of the [[group]] set such that every element of the [[group]] can be expressed as a combination (under the [[binary operation|group operation]]) of finitely many elements of the subset and their inverses. [^word] Explicitly, $A \subset G$ **generates $G$** if there exists $n$ and $g_{1},\dots,g_{n} \in A \cup A^{-1}$ [^1] s.t. $g=g_{1}\dots g_{n}$. > \ > **Remark.** If $G$ is a [[group]] and $S \subset G$ is a subset of $G$ (not necessarily a [[subgroup]]), then the [[subgroup]] $\langle S \rangle$ **generated by** $S$ is (pretty much by definition) the smallest [[subgroup]] of $G$ containing $S$ (the intersection of all supergroups). > \ > We call a [[group]] $G$ **finitely generated** if there exists a finite subset $S \subset G$ such that $G=\langle S \rangle$. ^cb4f10 > [!equivalence] > In terms of [[universal property|universal properties]]: if $S \subset G$, we have a unique [[group homomorphism]] $\varphi_{S}:F(S) \to G$ by the [[free group|universal property of free groups]]. $\im \varphi_{S}$ is exactly the [[subgroup]] generated by $S$. ^equivalence > [!basicproperties] > $\alpha \in G$ generates $G$ iff the [[group homomorphism|homomorphism]] $n \mapsto g^{n}$ is [[surjection|surjective]]. > > $\to$. Suppose $\alpha$ generates $G$ (i.e., $G$ is [[cyclic group|cyclic]]). Let $g \in G$. $g$ equals a [[word on a set]] in $m$ copies of $\alpha$ and $k$ copies of $\alpha ^{-1}$. After cancellations, we have that $g=\alpha^{m-k}$ for some integer $n:=m-k \in \mathbb{Z}$. Thus [[surjection|surjectivity]] is shown > > $\leftarrow$. Fix $g \in G$ and suppose $n \mapsto g^{n}$ is [[surjection|surjective]]. Then any element in $G$ may be written as $g^{n}$ for some $n \in \mathbb{Z}$. This is a word in $g$; hence $g$ generates $G$. > > Let $\alpha,\beta \in G$ such that $\alpha \beta=\beta \alpha$. Then $f:\mathbb{Z}^{2} \to G$ s.t. $f(k,n)=\alpha^{n}\beta^{k}$ for all $k,n$, is a [[group homomorphism|homomorphism]]. > > > We have $\begin{align} > f\big( (k_{1},n_{1})+ (k_{2},n_{2})\big ) = & f\big( k_{1}+k_{2}, n_{1}+n_{2} \big) \\ > = & \alpha^{n_{1}+n_{2}} \beta^{k_{1}+k_{2}} \\ > = & \alpha^{n_{1}} \alpha^{n_{2}} \beta^{k_{1}} \beta^{k_{2}} \\ > = & \alpha^{n_{1}} \beta^{k_{1}} \alpha^{n_{2}} \beta^{k_{2}} \\ > = & f(k_{1},n_{1})f(k_{2},n_{2}) > \end{align}$ > as required. ---- #### [^1]: Where $A^{-1}:= \{ a^{-1} : a \in A \}$. [^word]: I.e., a [[word on a set|word]] on $A$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```