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→The Free Group
Let <math>a,a',b,b' \ </math> be "letters." We will form "words" by stringing finitely many of these letters together, subject to just a few simplification rules: <math>aa' = a'a = bb' = b'b= e \ </math>, where <math>e \ </math> is the identity, shorthand for the "word" of zero letters.
Let <math>a,a',b,b' \ </math> be "letters." We will form "words" by stringing finitely many of these letters together, subject to just a few simplification rules: <math>aa' = a'a = bb' = b'b= e \ </math>, where <math>e \ </math> is the identity, shorthand for the "word" of zero letters.
We state without proof that such words form a group under "concatenation," or stringing back to back, ie, <math>aab' * bab' = aab'bab' = aaab' \ </math>. By using exponents, ie, <math>aaa = a^3 \ </math>, each word has a unique spelling.
This is called <math>F_2 \ </math>, the free group on two generators. Any group structured like this, in which ''a'' and ''b'' have infinite order and are independent, (ie, no combination of ''a'' and ''b'' form ''e'') is this group. We can picture this group graphically, with ''a'' and ''b'' axes:
===Paradoxical Decomposition of the Free Group===
===Paradoxical Decomposition of the Free Group===