Free Group On Two Generators
The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a, a-1, b and b-1 such that no a appears directly next to an a-1 and no b appears directly next to a b-1. Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string. For instance: "abab-1a-1" concatenated with "abab-1a" yields "abab-1a-1abab-1a", which gets reduced to "abaab-1a". One can check that the set of those strings with this operation forms a group with neutral element the empty string ε := "". (Usually the quotation marks are left off, which is why you need the symbol ε!)
This is another infinite non-abelian group.
Free groups are important in algebraic topology; the free group in two generators is also used for a proof of the Banach–Tarski paradox.
Read more about this topic: Examples Of Groups
Famous quotes containing the words free and/or group:
“In a free society the state does not administer the affairs of men. It administers justice among men who conduct their own affairs.”
—Walter Lippmann (1889–1974)
“No other group in America has so had their identity socialized out of existence as have black women.... When black people are talked about the focus tends to be on black men; and when women are talked about the focus tends to be on white women.”
—bell hooks (b. c. 1955)