Mathematics 1214: Introduction to Group Theory
Homework exercise sheet 8
Due 12:50pm, Friday 26th March 2010
1. Let G be a group and let a, b ∈ G.
(a) Prove that if a, b ∈ G, then a = b ⇐⇒ ab−1 = e.
(b) Prove that G is an abelian group if and only if aba−1 b−1 = e for all a, b ∈ G.
2. [Optional question]
Let G be a group and let a ∈ G. Prove that for any integers n, m ∈ Z, we have
an am = an+m .
[Suggestion: fix m ∈ Z and show that this works for n = 0 and n = 1, then prove
it by induction on n for n ≥ 1. Finally, think about what happens when n < 0.]
3. Let G be a group and let a ∈ G. Prove that for any integers n, m ∈ Z, we have
(an )m = anm . [Suggestion: fix n ∈ Z and show that this works for m = 0, and then
prove it by induction on m for m > 0. Then consider what happens when m < 0.]
4. Disprove the following statements.
(a) If G is a group and a, b ∈ G and n ∈ Z, then (ab)n = an bn .
(b) If G is a group and a, b ∈ G and n ∈ Z, then (ab)n = bn an .
5. Let G be a group and let a ∈ G.
(a) Show that hai = ha−1 i.
(b) Deduce from (a) that o(a) = o(a−1 ).
6. For each element a in the group Z10 , compute o(a) and the cyclic subgroup hai.
n
7. Let Q = { m
: n, m ∈ Z, m 6= 0} denote the set of rational numbers. It is not hard
to show that (Q, +) is an abelian group. Prove that (Q, +) is not a cyclic group.
8. Let n, m ∈ Z. [As usual, Z denotes the group (Z, +).]
(a) Compute the cyclic subgroups hni and hmi, and show that
hni ⊆ hmi ⇐⇒ m n.
(b) Find a sequence of H1 , H2 , H3 , . . . of cyclic subgroups of Z such that
H1 ) H2 ) H3 ) . . . .