Algebra. Homework 2. Due on February 10, 2023.
1. Determine if the following sets and operations define a group. It yes, give a
proof that the properties in the definition of a group (associativity, existence
of the neutral element, and existence of inverses) hold; if no give an example
that (at least one of) these properties does not hold.
(a) (Z, −) the set of integers with subtraction operation;
(b) (R∗ , ·) the set of nonzero real numbers R∗ = {x ∈ R, x 6= 0} with multiplication operation;
2. Let G be a group and let a, b ∈ G. Which of the following is the inverse of
ab?
(a) a−1 b−1 ;
(b) b−1 a−1 ?
3. Let G be a group and let a, b, c ∈ G. Which of the following is the inverse of
abc?
(a) a−1 b−1 c−1 ;
(b) c−1 b−1 a−1 .
Explain why.
4. Determine all groups G, such that the set G has exactly 3 elements (in other
words, determine all group structures on the set of three elements).
5. Determine all the subgroups of the group S3 .
6. Give an example of a group homomorphism f : Z/2Z → S3 .
7. Give an example of a group homomorphism f : Z/3Z → S3 .
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