Word (group Theory) - Normal Forms

Normal Forms

A normal form for a group G with generating set S is a choice of one reduced word in S for each element of G. For example:

• The words 1, i, j, ij are a normal form for the Klein four-group.
• The words 1, r, r2, ..., rn-1, s, sr, ..., srn-1 are a normal form for the dihedral group Dih_n.
• The set of reduced words in S are a normal form for the free group over S.
• The set of words of the form xmyn for m,n ∈ Z are a normal form for the direct product of the cyclic groups 〈x〉 and 〈y〉.