---- > [!definition] Definition. ([[presentation of a group]]) > > Every [[group]] $G$ can be [[surjection|surjected]] on by a [[free group]] (e.g., by $F(G)$), implying by the corollary in [[first isomorphism theorem]] that every group is [[group isomorphism|isomorphic]] to a [[quotient group|quotient]] of a [[free group]]. > > A **presentation** of $G$ is an explicit such [[group isomorphism|isomorphism]] $G \cong \frac{F(A)}{R}$ where $A$ is a set and $R$ is a [[subgroup]] of 'relations'. In other words, a presentation is encoded by an explicit [[surjection]] $\rho: F(A) \to G$ > whose [[kernel of a group homomorphism|kernel]] is $R$. Usually $R$ is represented as a set of [[word on a set|words]] > > $\langle a,b,c \rangle/ \langle a + b - c \rangle = \langle a ,b ,c | a+b-c=0 \rangle $ > > ---- #### ---- #### 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 > ```