MATH 113 Lecture Notes - Lecture 19: Abelian Group, Free Group, Substring
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Fact: suppose a is an abelian group; suppose b, c are subgroups of a such that every a a can be uniquely expressed as a = b+c for b b and c c. Then, a is isomorphic to the cartesian product b x c. Defined by = = (b, c) Have to show: zxz is isomorphic to x <1, 1> Then (a, b) = a * (1, 0) + b * (0,1 ) where (0, 1) = (1, 1) (1, 0) This is equal to a(1, 0) + b(1, 1) b(1, 0) Suppose (a, b) = x(1, 0) + y(1, 1) We eventually get to prove that z x z is isomorphic to <1, 0> x <1, 1>, and if we divide both sides by <1, 1>, we get that zxz/<1, 1> is isomorphic to <1, 0>. Example: the free group on finitely many generators and its subgroups.
