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
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