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:
“When a man says that he is Jesus or Napoleon, or that the Martians are after him, or claims something else that seems outrageous to common sense, he is labeled psychotic and locked up in a madhouse. Freedom of speech is only for normal people.”
—Thomas Szasz (b. 1920)
“The government, which is the supreme authority in states, must be in the hands of one, or of a few, or of the many. The true forms of government, therefore, are those in which the one, the few, or the many, govern with a view to the common interest.”
—Aristotle (384–323 B.C.)