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 Dihn.
- 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〉.
Read more about this topic: Word (group Theory)
Famous quotes containing the words normal and/or forms:
“Cant is always rather nauseating; but before we condemn political hypocrisy, let us remember that it is the tribute paid by men of leather to men of God, and that the acting of the part of someone better than oneself may actually commit one to a course of behaviour perceptibly less evil than what would be normal and natural in an avowed cynic.”
—Aldous Huxley (1894–1963)
“From the war of nature, from famine and death, the most exalted object which we are capable of conceiving, namely, the production of the higher animals, directly follows. There is grandeur in this view of life, with its several powers, having been breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.”
—Charles Darwin (1809–1882)