Generators and relations

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 20 March 2012

Generators and relations

Let S be a set. Let A be an algebra.

A free A-module on S is a pair (AS,ι) where

  1. AS is an A-module
  2. ι:SAS is a function
such that,
  1. if M is an A-module with a function ȷ:SM then there exists a unique morphism
  2. ȷ˜: ASM such that ȷ˜ι = ȷ.
S AS M ι ȷ ȷ˜
This is a definition by a universal property.

The forgetful functor is F: A-modules Sets M M

The free module functor is Sets A-modules S AS

The universal property for free modules tells us HomA (AS,M) HomSets (S,M) ȷ˜ ȷ The free module functor is the left adjoint to the forgetful functor.

Let M be an A-module.

A presentation of M is an exact sequence AT ρ AS γ M0 where AT and AS are free modules.

The term exact sequence means that

  1. imρ=kerγ
  2. imγ=ker 0.

Let X be a group.

A presentation of X is an exact sequence RGX{1} where R and G are free groups.


  1. m is generated by 1 with relation 1+1++1 m   times =0. The cyclic group of order m Cm is generated by g with relation gm=1.

    Alternatively: m is presented by m 0 1 1 1 m and Cm is presented by G{r} G{g} Cm {1} g g r gm where GS denotes the free group on the set S.
  2. The dihedral group of order 2m, Gm,m,2 is generated by x and y with relations x2=1,  ym=1   and   xy=y-1x. Alternately: F3 F2 Gm,m,2 {1} g1 x g2 y r1 g12 r2 g2m r3 g1g2g1-1g2 where Fk denotes a free group on a set with k elements.

Presentations by exact sequences

A presentation of E is an exact sequence F(Y) F(X) E0 where F(Y) is the free object on the set Y and F(X) is the free object on the set X.

An exact sequence is a sequence of morphisms Ei γi Ei+1 γi+1 Ei+2 γi+2 such that if i then imγi = kerγi+1.

Let G be a group and let g1...gnG and r1r2...rm F{x1,...,xn}.

The group G is presented by generators and relations r1r2...rm if F{y1,...,ym} F{x1,...,xn} G {1} xi gi yj rj is an exact sequence.

Let V be a vector space and let b1,...,bn V.

The vector space V has basis {b1,...,bn} if {0} 𝔽{x1,...,xn} V {0} xi bi is an exact sequence.

  1. A cyclic group is a group generated by one element.
  2. A dihedral group is a group generated by two elements of order two.

Some "familiar" constructions by generators and relations

  1. 𝔽[x1,...,xn] is the free commutative algebra on the set {x1,...,xn}.
  2. The free group on the set X={x1,...,xn} is the monoid given by generators {x1,...,xn,y1,...,yn} and relations x1y1=1,   x2y2=1,   ...,   xnyn=1.
  3. Let V and W be R-modules, V a right R-module and W a left R-module. The tensor product is the abelian group generated by {vw  |  vV,  wW} with relations (z1v1+z2v2) w = z1(v1w) + z2(v2w) v (z1w1+z2w2) = z1(vw1) + z2(vw2) vrw = vrw for v,v1,v2V,   w,w1,w2W,   z1,z2,   rR.
  4. 𝔽[x1±1,...,xn±1] is the commutative algebra given by generators x1,...,xn,y1,...,yn and relations x1y1=1,   x2y2=1,   ...,   xnyn=1.
  5. The free abelian group n is the group given by generators x1,...,xn with relations xi+xj = xj+xi , for   i,j {1,2,...,n}.

The goal of classical invariant theory

Let V be a vector space with basis {x1,...,xn}. Then S(V) = 𝔽[x1,...,xn]. Let W0 be a subgroup of GL(V). Since S(V) is a functor, W0 acts on S(V): w(xi1xik) = (wxi1) (wxi2) (wxik) for wW0, 1i1ikn.

The invariant ring of W0 is S(V)W0 = {fS(V)  |  wf=f   for   wW0}.

Find generators and relations for S(V)W0.

Let V be a free -module with basis {x1,...,xn}. Then 𝔽[V] = 𝔽[x1±1,...,xn±1]. Let W0 be a subgroup of GL(V) (note that GL(V) = GLn() ), and wxλ = xwλ where xλ = x1λ1 xnλn if λ= λ1ε1 ++ λnεn .

The invariant ring of W0 is 𝔽[V]W0 = {f𝔽[V]  |  wf=f   for   wW0}.

  1. The cyclic groups are =G,1,1 ,   and   r =Gr,1,1.
  2. The dihedral groups are I2(r) = Gr,r,2   and   I2() = G,,2.

Notes and References

The definition of presentation is found in [Bou, Alg. Ch.10, §1 no.4 (13)]. The definition of cyclic group is in [Bou, Alg.] and the definition of dihedral group is from [Bou, Lie, Ch.IV, §1 Def.2].



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