GROUPS
5 minute review. Recap the four axioms for a group (reproduced on page 3 for
reference). Give a few examples of groups without fully justifying the axioms, such
as (Z, +), (C \ {0}, ×), Sn under composition, GL2 (F ) for F = R, Q, C or Zp .
Class warm-up. Let S be any set. The power set of S, denoted P(S), consists
of all the subsets of S. Notice that union and intersection are binary operations on
P(S): if A, B ∈ P(S) then A ∪ B ∈ P(S) and A ∩ B ∈ P(S).
For A ∈ P(S), simplify (i) A ∪ ∅, (ii) A ∩ ∅, (iii) A ∪ S and (iv) A ∩ S. Are there
neutral elements for ∪ and ∩ in P(S)? Do ∪ or ∩ turn P(S) into a group?
Problems. Choose from the below.
1. By constructing Cayley tables, decide whether the following are groups under
multiplication mod 10.
(a) G1 = {1, 3, 7, 9};
(b) G2 = {1, 2, 4, 6, 8}.
2. Which of 2 × 2 matrices below are members of GL2 (F ) for the given field F ?
For those that are, find their inverses.
1 2
(a) A1 =
with F = Z5 ;
3 4
2 1
(b) A2 =
with F = Z5 ;
1 3
0
p−1
with F = Zp , where p is prime;
(c) A3 =
p−1
0
i −1
(d) A4 =
with F = C.
1 1
3. Let G = {e, a, b, c} be a group of order 4, where e is the identity element.
Given that a2 = b, use the Latin square property to complete a Cayley table
for G.
4. Let S and P(S) be as in the warm-up. For A, B ∈ P(S), the symmetric
difference A∆B is defined by A∆B = (A ∪ B)\(A ∩ B), i.e. those elements
in A or B but not both.
(a) Let S = {1, 2}. Complete a Cayley table for P(S) = {∅, {1}, {2}, S}
under symmetric difference.
(b) For A ∈ P(S), simplify the expressions A∆∅ and A∆A.
(c) Given ∆ is associative, is P(S) a group under ∆?
5. By constructing a Cayley table, decide whether G3 = {2, 4, 6, 8} is a group
under multiplication mod 10. Does your answer unsettle you? Can you create
similar examples?
Homework. Chapter 3, Qs 4 & 5
1
2
GROUPS
For the warm-up, (i) A ∪ ∅ = A = ∅ ∪ A, (ii) A ∩ ∅ = ∅, (iii) A ∪ S = S and
(iv) A ∩ S = A = S ∩ A. It follows that ∅ is neutral for ∪ and S is neutral for ∩.
Neither turn P(S) into a group, though, due to the absence of inverses.
Selected answers and hints.
1. The set G1 is a group (closure, associativity, neutral elements and inverses all
present), but not G2 : 1 is a neutral element, but 2 has no inverse.
3 1
−1
2. (a) Matrix A1 is invertible, so is in GL2 (Z5 ), with A1 =
.
4 2
(b) As det A2 = 6 − 1 = 0, it follows that A2 is not invertible so is not in
GL2 (Z5 ).
2
(c) Using the fact that p − 1 = 1 in Zp , matrix A3 has determinant −(p − 1)2 =
−1 = p − 1, so is invertible (and hence a member of GL2 (Zp )) with
0
−(p − 1)
0 1
−1
A−1
=
p
−
1
=
p
−
1
= A3 .
3
−(p − 1)
0
1 0
1 1−i 1−i
−1
(d) Matrix A4 is also invertible as det A4 = 1+i, and A4 =
.
2 i−1 1+i
G
e
3. The completed table is a
b
c
e
e
a
b
c
a
a
b
c
e
P(S)
∅
4. (a) The completed table is {1}
{2}
S
b
b
c
e
a
c
c
e .
a
b
∅ {1} {2} S
∅ {1} {2} S
{1} ∅
S {2} .
{2} S
∅ {1}
S {2} {1} ∅
(b) We have A∆∅ = A and A∆A = ∅. Note that ∅∆A is also equal to A.
Hence ∅ is a neutral element for ∆. For every A ∈ P(S), A∆A = ∅ so
A is its own inverse. As P(S) is closed and ∆ is associative (given), it
follows that P(S) is a group under symmetric difference.
×mod10
2
5. Here, the completed table is
4
6
8
2
4
8
2
6
4
8
6
4
2
6
2
4
6
8
8
6
2 .
8
4
The strange thing here is that 6 is acting as a neutral element (look carefully!),
and every element has an inverse (e.g. 2 has inverse 8). Closure is also
apparent, and the operation is associative. That means G3 is a group! But
this is a bit unsettling, as we’d normally expect 1 to be the neutral element
for any group under modular multiplication. I guess it goes to show, we need
to be very careful about any assumptions we make!
For other examples of this kind of thing, try multiples of p1 under multiplication modulo p1 p2 , where p1 and p2 are prime.
For more details, start a thread on the discussion board.
GROUPS
3
Definition. A non-empty set G is a group under (more formally, (G, ) is a
group) if the following four axioms hold.
G1 (Closure): is a binary operation on G. That is, a b ∈ G for all a, b ∈ G.
G2 (Associativity): (a b) c = a (b c) for all a, b, c ∈ G.
G3 (Neutral element): There is an element e ∈ G such that, for all g ∈ G,
eg =g =ge
Such an element is called a neutral or identity element for G.
G4 (Inverses): For each element g ∈ G there is an element h ∈ G such that
g h = e = h g.
Such an element h is called an inverse of g.