Introduction to Abstract Algebra/Problem set 3

Derivations of Properties

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In the following exercises, you are prompted to give proofs which support the statements. Let   denote the identity element of some group.

  1. Prove that if   or  , then  .
  2. Prove that if  , then  .

In the following exercises, you are prompted for proofs supporting the statements regarding various subsets of the real numbers,  . For reference,   and  .

  1. Prove that   is a group where   for  .
  2. Prove that   does not form a group.
  3. Prove that a homomorphism from   to   exists.
  4. Prove that a homomorphism from   to   exists.
  5. Prove that there is a bijection,   for which it is true that   and  . We say that   is an isomorphism between the two groups.