Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 2
1. Let ∗ be an operation on a set S. Suppose that e is an identity element for ∗, and
that there are elements x, y1 , y2 ∈ S such that
x ∗ y1 = e = x ∗ y2 ,
y1 ∗ x = e = y2 ∗ x and y1 6= y2 .
(a) Explain why ∗ must fail to be associative.
(b) Find the Cayley table of an operation with these properties on S = {e, f, g}.
2. Let ∗ be an associative operation on a set S, and suppose that e is an identity
element for ∗. Show that if x, y ∈ S with
x ∗ y = e and y ∗ x 6= e
then x is not invertible with respect to ∗. [Hint: try a proof by contradiction.]
3. Let ∗ be an operation on a set S with an identity element e ∈ S, and let us write
T = {x ∈ S : x 6= e}. Prove that ∗ is associative if and only if
x ∗ (y ∗ z) = (x ∗ y) ∗ z
for all x, y, z ∈ T .
[Note: the statement “x, y, z ∈ T ” is shorthand for “x ∈ T , y ∈ T and z ∈ T ”.]
4. Let ∗ be an operation on a set S, and let ~ : S × S → S be given by
x~y =y∗x
(a)
(b)
(c)
(d)
for x, y ∈ S.
Explain how the Cayley tables of ∗ and ~ are related.
Prove that ∗ is commutative ⇐⇒ ∗ = ~ ⇐⇒ ~ is commutative.
If (S, ∗) is a group, prove that (S, ~) is a group.
Is the converse of (c) true? Either give a proof or a counterexample.
[The converse of the statement “if P then Q” is “if Q then P ”.]
5. Given a group (G, ∗) and x, y ∈ G, consider the following two mappings:
Lx : G → G, Lx (z) = x ∗ z
and Ry : G → G, Ry (z) = z ∗ y
for z ∈ G.
(a) Prove that Lx ◦ Ry = Ry ◦ Lx .
(b) Prove that (G, ∗) is abelian if and only if Lx = Rx for every x ∈ X.
(c) Prove that the following statements are equivalent:
(i). x = y
(ii). Lx = Ly
(iii). Rx = Ry
[This means: show that any of these three statements is true if and only if
any of the other statements is true. One way to do this is to prove that
(i) ⇐⇒ (ii), and that (i) ⇐⇒ (iii).]
6. Which of the following is a group operation on M (2, R), the set of 2 × 2 matrices
with real entries? As always, you should prove that your answers are correct.
(a) matrix addition
(b) matrix multiplication
(c) the operation ∗ given by A ∗ B = AB − BA for matrices A and B