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Group Presentation

A group specified as a quotient of a free group by the normal closure of a set of relations.

A group presentation consists of a set SS of generators and a set R⊆F(S)R\subseteq F(S) of relators. The notation

⟨S∣R⟩\langle S \mid R\rangle

denotes the quotient of the F(S)F(S) by the of RR:

⟨S∣R⟩  :=  F(S)/ ⁣⟨ ⁣⟨R⟩ ⁣⟩,\langle S \mid R\rangle \;:=\; F(S)\big/\!\langle\!\langle R\rangle\!\rangle,

a in which every relator is equal to the identity.

Remarks

A presentation is finite when both SS and RR are finite. Different presentations can define isomorphic groups.

Examples
  • The cyclic group of order nn has presentation ⟨a∣an=e⟩\langle a\mid a^n=e\rangle.
  • The free abelian group of rank 22 has presentation ⟨a,b∣aba−1b−1=e⟩\langle a,b\mid aba^{-1}b^{-1}=e\rangle.
  • The dihedral group D2nD_{2n} has presentation ⟨r,s∣rn=e, s2=e, srs−1=r−1⟩\langle r,s\mid r^n=e,\ s^2=e,\ srs^{-1}=r^{-1}\rangle.