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Group Presentation

A description of a group by generators and relations
Group Presentation

A group presentation is a way to specify a by choosing a SS and a set of relations RR among those generators. Formally, the presentation

⟨S∣R⟩ \langle S \mid R\rangle

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

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

a . Intuitively, one starts with all formal words in SS and forces the relations in RR to hold.

Presentations are ubiquitous: they encode groups by finitely many symbols when possible, and many structural questions reduce to understanding the relations.

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∣ab=ba⟩\langle a,b \mid ab=ba\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.