Suggested Exercises I
Symmetry & Groups
1. Write down all multiplication table for S3 and all elements of S4 .
2. Show a group of order 5 must be abelian.
3. Determine if the set G a set with more than one element is a group with multiplication defined by
a ∗ b = a for all a, b ∈ G.
4. Let G be a set with operation ∗ such that:
(a) G is closed under ∗,
(b) ∗ is associative,
(c) there exists an element e ∈ G such that e ∗ x = x for all x ∈ G, and
(d) given x ∈ G, there exists a y ∈ G such that y ∗ x = e.
Prove that G is a group.
5. Give a counter example in GL2 (R) to the cancellation lemma.
6. Give an example of a group homomorphism which is:
(a) injective but not surjective,
(b) surjective but not injective,
(c) neither injective nor surjective,
(d) an isomorphism.
1