free - theory(basically that a symbol and its inverse next...

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A (nontrivial) cautionary exercise about the proper use of letters Let G be a group (with the usual multiplicative notation). An element of the form a - 1 b - 1 ab is called a commutator of G . The subgroup G 0 generated by all commutators is called the commutator subgroup of G : G 0 = h{ a - 1 b - 1 ab : a, b G }i (Remark: This is a sophisticated subgroup with lots of useful properties.) Let G = F ( a, b ), the free group generated by the two distinct symbols a and b . You can think of the elements of G as strings consisting of the symbols a, b, a - 1 , b - 1 (the empty string being the identity of the group), with concatenation as multiplication, the only relations being the axioms of group
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Unformatted text preview: theory (basically, that a symbol and its inverse next to each other cancel out). For a precise list of relations, see below. Prove that the commutator subgroup of G is the set of strings in which there are exactly as many ‘ a ’-s as ‘ a-1 ’-s, and there are exactly as many ‘ b ’-s as ‘ b-1 ’-s (including among them the identity itself). List of relations in F ( a, b ) (writing e for the identity) ( ∀ a ∈ G )[ ea = ae = a ] ( ∀ a ∈ G )[ aa-1 = a-1 a = e ] ( ∀ b ∈ G )[ bb-1 = b-1 b = e ]...
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