A note on finitely generated/related/presented groups

Group presentation is a method to express groups as a quotient of a free group and a normal subgroup. In a presentation, there are generators and relators, and the presentation yields a ‘minimalist implementation’ of these generators and relations. Some cases of special interests are finitely generated/related/presented groups. From the definition, finite generation + finite relation is not linguistically equivalent to finite presentation. There has been some discussion on the relation of the three properties for several other algebraic structures. This entry proves that finite generation + finite relation imples finite presentation for groups.

For modules, finite generation and finite relation implies finite presentation. For abelian groups (i.e., Z\mathbb{Z}-modules), finite generation implies finite relation and finite presentation.

We’ll go through the necessary definitions and then prove the following proposition:

If SRSR\lAng{S|R'}\rAng\cong\lAng{S'|R}\rAng for some finite S,RS,R, then both are isomorphic to SR\lAng{S''|R''}\rAng with finite S,RS'',R''. That is, a group that is both finitely generated and finitely related must be finitely presented.

Definitions

Group, free group (constructive, F(S)F\left(S\right)), generated (normal) subgroup (S\lAng S\rAng and SN{\lAng S\rAng}_N) are omitted.

Presentation Given a pair of set S,RS,R, where RF(S)R\subseteq F\left(S\right), the group F(S)/RN{F\left(S\right)}/{{\lAng R\rAng}_N} is denoted by SR\lAng{S|R}\rAng.

Finite -tion If a group GG is isomorphic to some SR\lAng{S|R}\rAng where SS [resp. RR, both SS and RR] is [resp. is, are] finite, GG is said to be finitely generated [resp. related, presented].

Remarks Here the definition of finitely generated group is different from what is usually known in the textbooks. One can regard the textbook definition as the internal version and the one here as the external version. Indeed, they are equivalent.

If G=SG=\lAng S\rAng for some finite SGS\subseteq G, define R={s1s2sn:s1s2sn=1,siSsi1S,nN},R=\left\{s_1s_2\cdots s_n:s_1s_2\cdots s_n=1,s_i\in S\vee s_i^{-1}\in S,n\in\mathbb{N}\right\}, then GSRG\cong\lAng{S|R}\rAng.

Conversely, if SRG\lAng{S|R}\rAng\cong G for some finite SS via isomorphism φ\varphi, then G={φ(sRN):sS}.G=\lAng\left\{{\varphi\left(s{\lAng R\rAng}_N\right):s\in S}\right\}\rAng.

Proof

We now prove that if a group is both finitely generated and finitely related, it is finitely presented.

Suppose SRSR\lAng{S|R'}\rAng\cong\lAng{S'|R}\rAng where S,RS,R are finite. Let φ:SRSR\varphi:\lAng{S|R'}\rAng\to\lAng{S'|R}\rAng be an isomorphism. Write N=RNN={\lAng R\rAng}_N and N=RNN'={\lAng R'\rAng}_N.

We write φ(sN)=sN\varphi\left(sN\right)=s'N' for each sSs\in S, where ss' is a finite product of elements and their inverses in SS'. I.e., they use finitely many symbols from SS'. For each relator in RR, it uses only finitely many symbols from SS'. Collect these symbols and put them into a (finite) set TT.

We claim S=TS'=T. Suppose not, and sSTs'\in{S'\setminus T}. Define f:SZ,xI[x=s],f:S'\to\mathbb{Z},x\mapsto{\mathbb{I}\left[{x=s'}\right]}, where I\mathbb{I} is the truth indication function, i.e., ff maps ss' to 11 and everything else in SS' to 00. (Uniquely) extend ff to a homomorphism ψ:F(S)Z\psi:F\left(S'\right)\to\mathbb{Z}. Note that for all tTt\in T, by definition, ψ(t)=f(t)=0\psi\left(t\right)=f\left(t\right)=0, and since RR only uses symbols in TT, we conclude NkerψN\leq\ker\psi. Now, sN=φ(φ1(sN))=tN,s'N=\varphi\left({\varphi^{-1}\left(s'N\right)}\right)=tN, where tt is a finite product of elements and their inverses in TT. To get this, first write φ1(sN)\varphi^{-1}\left(s'N\right) as a coset of NN' translated by a finite product of elements and their inverses in SS. Then apply φ\varphi on each term in the product, and write the outcome as a coset of NN translated by a product of finite products of elements and their inverses in TT (by our choice). Applying ψ\psi to both sides of the equation yields 1=01=0, a contradiction.

Therefore, S=TS'=T and SR\lAng{S'|R}\rAng is a finite presentation.

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