Group theory notes

M2PM2 Notes on Group Theory
Here are some notes on the M2PM2 lectures on group theory.
1 Revision from M1P2
Would be a good idea to refresh your memory on the following topics from
group theory.
(a) Group axioms: closure, associativity, identity, inverses
(b) Examples of groups:
(Z, +), (Q, +), (Q∗, ×), (C∗, ×), etc
GL(n, R), the group of all invertible n×n matrices over R, under matrix
multiplication
Sn, the symmetric group, the set of all permutations of {1, 2, . . . , n},
under composition. Recall the cycle notation for permutations – every per-
mutation can be expressed as a product of disjoint cycles.
For p prime Z∗
p = {[1], [2], . . . , [p − 1]} is a group under multiplication
modulo p.
Cn = {x ∈ C : xn = 1} = {1, ω, ω2, . . . , ωn−1} is a cyclic group of size n,
where ω = e2πi/n.
(c) Some theory:
Criterion for subgroups: H is a subgroup of G iff (1) e ∈ H; (2) x, y ∈
H ⇒ xy ∈ H, and (3) x ∈ H ⇒ x−1 ∈ H.
For a ∈ G, we define the cyclic subgroup a = {an : n ∈ Z}. The size
of a is equal to o(a), the order of a, which is defined to be the smallest
positive integer k such that ak = e.
Lagrange: if H is a subgroup of a finite group G then |H| divides |G|.
Consequences: (1) For any element a ∈ G, o(a) divides |G|.
(2) If |G| = n then xn = e for all x ∈ G
(3) If |G| is prime then G is a cyclic group.
1

2 More examples: symmetry groups
For any object in the plane R2 (later R3) we’ll show how to define a group
called the symmetry group of the object. This group will consist of functions
called isometries, which we now define. Recall for x = (x1, x2), y = (y1, y2) ∈
R2, the distance
d(x, y) = (x1 − y1)2 + (x2 − y2)2.
We define an isometry of R2 to be a bijection f : R2 → R2 which preserves
distance, i.e. for all x, y ∈ R2,
d(f(x), f(y)) = d(x, y).
There are many familiar examples of isometries:
(1) Rotations: let ρP,θ be the function R2 → R2 which rotates every
point about P through angle θ. This is an isometry.
(2) Reflections: if l is a line, let σl be the function which sends every
point to its reflection in l. This is an isometry.
(3) Translations: for a ∈ R2, let τa be the translation sending x → x + a
for all x ∈ R2. This is an isometry.
Not every isometry is one of these three types – for example a glide-reflection
(i.e. a function of the form σl ◦τa) is not a rotation, reflection or translation.
Define I(R2) to be the set of all isometries of R2. For isometries f, g, we
have the usual composition function f ◦ g defined by f ◦ g(x) = f(g(x)).
Proposition 2.1 I(R2) is a group under composition.
Proof Closure: Let f, g ∈ I(R2). We must show f ◦ g is an isometry. It is
a bijection as f, g are bijections (recall M1F). And it preserves distance as
d(f ◦ g(x), f ◦ g(y)) = d(f(g(x)), f(g(y)))
= d(g(x), g(y) (as f is isometry)
= d(x, y) (as g is isometry).
Assoc: this is always true for composition of functions (since f ◦(g ◦h)(x) =
(f ◦ g) ◦ h(x) = f(g(h(x)))).
Identity is the identity function e defined by e(x) = x for all x ∈ R2, which
is obviously an isometry.
2

Inverses: let f ∈ I(R2). Then f−1 exists as f is a bijection, and f−1
preserves distance since
d(f−1
(x), f−1
(y)) = d(f(f−1
(x)), f(f−1
(y))) = d(x, y).
So we’ve checked all the axioms and I(R2) is a group.
Now let Π be a subset of R2. For a function g : R2 → R2,
g(Π) = {g(x) | x ∈ Π}
Example: Π =square with centre in the origin and aligned with axes,
g = ρπ/4. Then g(Π) is the original square rotated by π/4.
Definition The symmetry group of Π is G(Π) – the set of isometries g such
that g(Π) = Π, i.e.
G(Π) = g ∈ I(R2
) | g(Π) = Π .
Example: For the square from the previous example, G(Π) contains ρπ/2,
σx. . .
Proposition 2.2 G(Π) is a subgroup of I(R2).
Proof We check the subgroup criteria:
(1) e ∈ G(Π) as e(Π) = Π.
(2) Let f, g ∈ G(Π), so f(Π) = g(Π) = Π. So
f ◦ g(Π) = f(g(Π)) (1)
= f(Π) (2)
= Π. (3)
So f ◦ g ∈ G(Π).
(3) Let f ∈ G(Π), so
f(Π) = Π.
Apply f−1 to get
f−1
(f(Π)) = f−1
(Π) (4)
Π = f−1
(Π) (5)
and f−1 ∈ G(Π).
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So we have a vast collection of new examples of groups G(Π).
Examples
1. Equilateral triangle (= Π)
Here G(Π) contains
3 rotations: e = ρ0, ρ = ρ2π/3, ρ2 = ρ4π/3,
3 reflections: σ1 = σl1 , σ2 = σl2 , σ3 = σl3 .
Each of these corresponds to a permutation of the corners 1, 2, 3:
e ∼ e, (6)
ρ ∼ (1 2 3), (7)
ρ2
∼ (1 3 2), (8)
σ1 ∼ (2 3), (9)
σ2 ∼ (1 3), (10)
σ3 ∼ (1 2). (11)
Any isometry in G(Π) permutes the corners. Since all the permu-
tations of the corners are already present, there can’t be any more
isometries in G(Π). So the Symmetry group of equilateral triangle is
e, ρ, ρ2
, σ1, σ2, σ3 ,
called the dihedral group D6.
Note that it is easy to work out products in D6: e.g.
ρσ3 ∼ (1 2 3)(1 2) = (1 3) (12)
∼ σ2. (13)
2. The square
Here G = G(Π) contains
4 rotations: e, ρ, ρ2, ρ3 where ρ = ρπ/2,
4 reflections: σ1, σ2, σ3, σ4 where σi = σli
.
So |G| ≥ 8. We claim that |G| = 8: Any g ∈ G permutes the corners
1, 2, 3, 4 (as g preserves distance). So g sends
1 → i, (4 choices of i)
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2 → j, neighbour of i, (2 choices for j)
3 → oppositeofi,
4 → oppositeofj.
So |G| ≤ (num. of choices for i) × (for j) = 4 × 2 = 8. So |G| = 8.
Symmetry group of the square is
e, ρ, ρ2
, ρ3
, σ1, σ2, σ3, σ4 ,
called the dihedral group D8.
Can work out products using the corresponding permutations of the
corners.
e ∼ e, (14)
ρ ∼ (1 2 3 4), (15)
ρ2
∼ (1 3)(2 4), (16)
ρ3
∼ (1 4 3 2), (17)
σ1 ∼ (1 4)(2 3), (18)
σ2 ∼ (1 3), (19)
σ3 ∼ (1 2)(3 4), (20)
σ4 ∼ (2 4). (21)
For example
ρ3
σ1 → (1 4 3 2)(1 4)(2 3) = (1 3) (22)
→ σ2. (23)
Note that not all permutations of the corners are present in D8, e.g.
(1 2).
More on D8: Define H to be the cyclic subgroup of D8 generated by
ρ, so
H = ρ = e, ρ, ρ2
, ρ3
.
Write σ = σ1. The right coset
Hσ = σ, ρσ, ρ2
σ, ρ3
σ
is different from H.
H Hσ
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So the two distinct right cosets of H in D8 are H and Hσ, and
D8 = H ∪ Hσ.
Hence
Hσ = ρ, ρσ, ρ2
σ, ρ3
σ (24)
= {σ1, σ2, σ3, σ4} . (25)
So the elements of D8 are
e, ρ, ρ2
, ρ3
, σ, ρσ, ρ2
σ, ρ3
σ.
To work out products, use the “magic equation” (see Sheet 1, Question
2)
σρ = ρ−1
σ.
3. Regular n-gon
Let Π be the regular polygon with n sides. Symmetry group G = G(Π)
contains
n rotations: e, ρ, ρ2, . . . , ρn−1 where ρ = ρ2π/n,
n reflections σ1, σ2, . . . , σn where σi = σli
.
So |G| ≥ 2n. We claim that |G| = 2n.
Any g ∈ G sends corners to corners, say
1 → i, (n choices for i)
2 → j neighbour of i. (2 choices for j)
Then g sends n to the other neighbour of i and n − 1 to the remaining
neighbour of g(n) and so on. So once i, j are known, there is only one
possibility for g. Hence
|G| ≤ number of choices for i, j = 2n.
Therefore |G| = 2n.
Symmetry group of regular n-gon is
D2n = e, ρ, ρ2
, . . . , ρn
, σ1, . . . , σn ,
the dihedral group of size 2n.
Again can work in D2n using permutations
ρ → (1 2 3 ∙ ∙ ∙ n) (26)
σ1 → (2 n)(3 n − 1) ∙ ∙ ∙ (27)
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4. Benzene molecule
C6H6. Symmetry group is D12.
5. Infinite strip of F’s
. . . F F F . . .
−1 0 1
What is symmetry group G(Π)?
G(Π) contains translation
τ(1,0) : v → v + (1, 0).
Write τ = τ(1,0). Then G(Π) contains all translations τn = τ(n,0). Note
G(Π) is infinite. We claim that
G(Π) = {τn
| n ∈ Z} (28)
= τ , (29)
infinite cyclic group.
Let g ∈ G(Π). Must show that g = τn for some n. Say g sends F at 0
to F at n. Note that τ−n sends F at n to F at 0. So τ−ng sends F at 0
to F at 0. So τ−ng is a symmetry of the F at 0. It is easy to observe
that F has only symmetry e. Hence
τ−n
g = e (30)
τn
τ−n
g = τn
(31)
g = τn
. (32)
Note Various other figures have more interesting symmetry groups, e.g.
infinite strip of E’s, square tiling of a plane, octagons and squares tiling of
the plane, 3 dimensions – platonic solids. . . later.
3 Isomorphism
Let G = C2 = {1, −1}, H = S2 = {e, a} (where a = (1 2)). Multiplication
tables:
Of G : 1 −1
1 1 −1
−1 −1 1
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Of H : e a
e e a
a a e
These are the same, except that the elements have different labels (1 ∼ e,
−1 ∼ a).
Similarly for G = C3 = {1, ω, ω2}, H = a = {e, a, a2} (where a =
(1 2 3) ∈ S3):
Of G : 1 ω ω2
1 1 ω ω2
ω ω ω2 1
ω2 ω2 1 ω
Of H : e a a2
e e a a2
a a a2 e
a2 a2 e a
Again, these are same groups with relabelling
1 ∼ e,
ω ∼ a,
ω2 ∼ a2.
In these examples, there is a “relabelling” function φ : G → H such that if
g1 → h1,
g2 → h2,
then
g1g2 → h1h2.
Definition G, H groups. A function φ : G → H is an isomorphism if
(1) φ is a bijection,
(2) φ(g1)φ(g2) = φ(g1g2) for all g1, g2 ∈ G.
If there exists an isomorphism φ : G → H, we say G is isomorphic to H
and write G ∼= H.
Notes 1. If G ∼= H then |G| = |H| (as φ is a bijection).
2. The relation ∼= is an equivalence relation, i.e.
• G ∼= G ,
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• G ∼= H ⇒ H ∼= G,
• G ∼= H, H ∼= K ⇒ G ∼= K.
Example Which pairs of the following groups are isomorphic?
G1 = C4 = i = {1, −1, i, −i} ,
G2 = symmetry group of a rectangle = {e, ρπ, σ1, σ2} ,
G3 = cyclic subgroup of D8 ρ = e, ρ, ρ2, ρ3 .
1. G1
∼= G3? To prove this, define φ : G1 → G2
i → ρ,
−1 → ρ2,
−i → ρ3,
1 → e,
i.e. φ : in → ρn. To check that φ is an isomorphism
(1) φ is a bijection,
(2) for m, n ∈ Z
φ(imin) = φ(im+n)
= ρm+n
= ρmρn
= φ(im)φ(in).
So φ is an isomorphism and G1
∼= G3.
Note that there exist many bijections G1 → G3 which are not isomor-
phisms.
2. G2
∼= G3 or G2
∼= G1? Answer: G2
∼= G1. By contradiction. Assume
there exists an isomorphism φ : G1 → G2. Say φ(i) = x ∈ G2, φ(1) = y ∈
G2. Then
φ(−1) = φ(i2
) = φ(i ∙ i) = φ(i)φ(i) = x2
= e
as g2 = e for all g ∈ G2. Similarly φ(1) = φ(1 ∙ 1) = φ(1)φ(1) = y2 = e. So
φ(−1) = φ(1), a contradiction as φ is a bijection.
In general, to decide whether two groups G, H are isomorphic:
• If you think G ∼= H, try to define an isomorphism φ : G → H.
• If you think G ∼= H, try to use the following proposition.
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Proposition 3.1 Let G, H be groups.
(1) If |G| = |H| then G ∼= H.
(2) If G is abelian and H is not abelian, then G ∼= H.
(3) If there is an integer k such that G and H have different number of
elements of order k, then G ∼= H.
Proof (1) Obvious.
(2) We show that if G is abelian and G ∼= H, then H is abelian (this
gives (2)). Suppose G is abelian and φ : G → H is an isomorphism. Let
h1, h2 ∈ H. As φ is a bijection, there exist g1, g2 ∈ G such that h1 = φ(g1)
and h2 = φ(g2). So
h2h1 = φ(g2)φ(g1)
= φ(g2g1)
= φ(g1)φ(g2)
= h1h2.
(3) Let
Gk = {g ∈ G | o(g) = k} ,
Hk = {h ∈ H | o(h) = k} .
We show that G ∼= H implies |Gk| = |Hk| for all k (this gives (3)).
Suppose G ∼= H and let φ : G → H be an isomorphism. We show that
φ sends Gk to Hk: Let g ∈ Gk, so o(g) = k, i.e.
gk
= eG, and gi
= eG for 1 ≤ i ≤ k − 1.
Now φ(eG) = eH, since
φ(eG) = φ(eGeG)
= φ(eG)φ(eG)
φ(eG)−1φ(eG) = φ(eG)
eH = φ(eG).
Also
φ(gi) = φ(gg ∙ ∙ ∙ g) (i times)
= φ(g)φ(g) ∙ ∙ ∙ φ(g)
= φ(g)i.
Hence
φ(g)k = φ(eG) = eH,
φ(g)i = eH for 1 ≤ i ≤ k − 1.
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In other words, φ(g) has order k, so φ(g) ∈ Hk. So φ sends Gk to Hk. As φ
is 1-1, this implies |Gk| ≤ |Hk|.
Also φ−1 : H → G is an isomorphism and similarly sends Hk to Gk,
hence |Hk| ≤ |Gk|. Therefore |Gk| = |Hk|.
Examples 1. Let G = S4, H = D8. Then |G| = 24, |H| = 8, so G ∼= H.
2. Let G = S3, H = C6. Then G is non-abelian, H is abelian, so G ∼= H.
3. Let G = C4, H = symmetry group of the rectangle = {e, ρπ, σ1, σ2}.
Then G has 1 element of order 2, H has 3 elements of order 2, so G ∼= H.
4. Question: (R, +) ∼= (R∗, ×)? Answer: No, since (R, +) has 0 elements
of order 2, (R∗, ×) has 1 element of order 2.
Cyclic groups
Proposition 3.2 (1) If G is a cyclic group of size n, then G ∼= Cn.
(2) If G is an infinite cyclic group, then G ∼= (Z, +).
Proof (1) Let G = x , |G| = n, so o(x) = n and therefore
G = e, x, x2
, . . . , xn−1
.
Recall
Cn = 1, ω, ω2
, . . . , ωn−1
,
where ω = e2πi/n. Define φ : G → G by φ(xr) = ωr for all r. Then φ is a
bijection, and
φ(xrxs) = φ(xr+s)
= ωr+s
= ωrωs
= φ(xr)φ(xs).
So φ is an isomorphism, and G ∼= Cn.
(2) Let G = x be infinite cyclic, so o(x) = ∞ and
G = . . . , x−2
, x−1
, e, x, x2
, x3
, . . . ,
all distinct. Define φ : G → (Z, +) by φ(xr) = r for all r. Then φ is an
isomorphism, so G ∼= (Z, +).
This proposition says that if we think of isomorphic groups as being
“the same”, then there is only one cyclic group of each size. We say: “up
to isomorphism”, the only cyclic groups are Cn and (Z, +).
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Example Cyclic subgroup 3 of (Z, +) is {3n | n ∈ Z}, infinite, so by the
proposition 3 ∼= (Z, +).
4 Even and odd permutations
We’ll classify each permutation in Sn as either “even” or “odd” (reason given
later).
Example For n = 3. Consider the expression
Δ = (x1 − x2)(x1 − x3)(x2 − x3),
a polynomial in 3 variables x1, x2, x3. Take each permutation in S3 to
permute x1, x2, x3 in the same way it permutes 1, 2, 3. Then each g ∈ S3
sends Δ to ±Δ. For example
for e, (1 2 3), (1 3 2) : Δ → +Δ,
for (1 2), (1 3), (2 3) : Δ → −Δ.
Generalizing this: for arbitrary n ≥ 2, define
Δ =
i<j
(xi − xj) ,
a polynomial in n variables x1, . . . , xn.
If we let each permutation g ∈ Sn permute the variables x1, . . . , xn just
as it permutes 1, . . . , n then g sends Δ to ±Δ.
Definition For g ∈ Sn, define the signature sgn(g) to be +1 if g(Δ) = Δ
and −1 if g(Δ) = −Δ. So
g(Δ) = sgn(g)Δ.
The function sgn : Sn → {+1, −1} is the signature function on Sn. Call g
an even permutation if sgn(g) = 1, and odd permutation if sgn(g) = −1.
Example In S3 e, (1 2 3), (1 3 2) are even and (1 2), (1 3), (2 3) are odd.
Given (1 2 3 5)(6 7 9)(8 4 10) ∈ S10, what’s its signature ? Our next
aim is to be able answer such questions instantaneously. This is the key:
Proposition 4.1 (a) sgn(xy) = sgn(x)sgn(y) for all x, y ∈ Sn
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(b) sgn(e) = 1, sgn(x−1) = sgn(x).
(c) If t = (i j) is a 2-cycle then sgn(t) = −1.
Proof (a) By definition
x(Δ) = sgn(x)Δ,
y(Δ) = sgn(y)Δ.
So
xy(Δ) = x(y(Δ))
= x(sgn(y)Δ)
= sgn(y)x(Δ) = sgn(y)sgn(x)Δ.
Hence
sgn(xy) = sgn(x)sgn(y).
(b) We have e(Δ) = Δ, so sgn(e) = 1. So
1 = sgn(e) = sgn(xx−1)
= sgn(x)sgn(x−1) (by (a))
and hence sgn(x) = sgn(x−1).
(c) Let t = (i j), i < j. We count the number of brackets in Δ that are
sent to brackets (xr − xs), r > s. These are
(xi − xj),
(xi − xi+1), . . . , (xi − xj−1),
(xi+1 − xj), . . . , (xj−1 − xj).
Total number of these is 2(j − i − 1)+1, an odd number. Hence t(Δ) = −Δ
and sgn(t) = −1.
To work out sgn(x), x ∈ Sn here’s what we shall do:
• express x as a product of 2-cycles
• use proposition 4.1
Proposition 4.2 Let c = (a1a2 . . . ar), an r-cycle. Then c can be expressed
as a product of (r − 1) 2-cycles.
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Proof Consider the product
(a1ar)(a1ar−1) ∙ ∙ ∙ (a1a3)(a1a2).
This product sends
a1 → a2 → a3 → ∙ ∙ ∙ → ar−1 → a1.
Hence the product is equal to c.
Corollary 4.3 The signature of an r-cycle is (−1)r−1.
Proof Follows from previous two props.
Corollary 4.4 Every x ∈ Sn can be expressed as a product of 2-cycles.
Proof From first year, we know that
x = c1 ∙ ∙ ∙ cm,
a product of disjoint cycles ci. Each ci is a product of 2-cycles by 4.2. Hence
so is x.
Proposition 4.5 Let x = c1 ∙ ∙ ∙ cm a product of disjoint cycles c1, . . . , cm of
lengths r1, . . . , rm. Then
sgn(x) = (−1)r1−1
∙ ∙ ∙ (−1)rm−1
.
Proof We have
sgn(x) = sgn(c1) ∙ ∙ ∙ sgn(cm) by 4.1(a)
= (−1)r1−1 ∙ ∙ ∙ (−1)rm−1 by 4.3.
Example (1 2 5 7)(3 4 6)(8 9)(10 12 83)(79 11 26 15) has sgn = −1.
Importance of signature
1. We’ll use it to define a new family of groups below.
2. Fundamental in the theory of determinants (later).
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Definition Define
An = {x ∈ Sn | sgn(x) = 1} ,
the set of even permutations in Sn. Call An the alternating group (after
showing that it is a group).
Theorem 4.6 An is a subgroup of Sn, of size 1
2 n!.
Proof (a) An is a subgroup:
(1) e ∈ An as sgn(e) = 1.
(2) for x, y ∈ An,
sgn(x) = sgn(y) = 1,
sgn(xy) = sgn(x)sgn(y) = 1,
so xy ∈ An,
(3) for x ∈ An, we have sgn(x) = 1, so by 4.1(b), sgn(x−1) = 1, i.e.
x−1 ∈ An.
(b) |An| = 1
2 n!: Recall that there are right cosets of An,
An = Ane, An(1 2) = {x(1 2) | x ∈ An} .
These cosets are distinct (as (1 2) ∈ An(1 2) but (1 2) /∈ An), and have equal
size (i.e. |An| = |An(1 2)|). We show that Sn = An ∪ An(1 2): Let g ∈ Sn.
If g is even, then g ∈ An. If g is odd, then g(1 2) is even (as sgn(g(1 2)) =
sgn(g)sgn(1 2) = 1), so g(1 2) = x ∈ An. Then g = x(1 2) ∈ An(1 2).
So |An| = 1
2 |Sn| = 1
2 n!.
Examples
1. A3 = {e, (1 2 3), (1 3 2)}, size 3 = 1
2 3!.
2. A4:
cycle shape e (2) (3) (4) (2, 2)
in A4? yes no yes no yes
no. 1 8 3
Total |A4| = 12 = 1
2 4!.
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3. A5:
cycle shape e (2) (3) (4) (5) (2, 2) (3, 2)
in A5? yes no yes no yes yes no
no. 1 20 24 15
Total |A5| = 60 = 1
2 5!.
5 Direct Products
So far, we’ve seen the following examples of finite groups: Cn, D2n, Sn, An.
We’ll get many more using the following construction.
Recall: if T1, T2, . . . , Tn are sets, the Cartesian product T1 ×T2 ×∙ ∙ ∙×Tn
is the set consisting of all n-tuples (t1, t2, . . . , tn) with ti ∈ Ti.
Now let G1, G2, . . . , Gn be groups. Form the Cartesian product G1 ×
G2 × ∙ ∙ ∙ × Gn and define multiplication on this set by
(x1, . . . , xn)(y1, . . . , yn) = (x1y1, . . . , xnyn)
for xi, yi ∈ Gi.
Definition Call G1 × ∙ ∙ ∙ × Gn the direct product of the groups G1, . . . , Gn.
Proposition 5.1 Under above defined multiplication, G1 × ∙ ∙ ∙ × Gn is a
group.
Proof
• Closure True by closure in each Gi.
• Associativity Using associativity in each Gi,
[(x1, . . . , xn)(y1, . . . , yn)] (z1, . . . , zn) = (x1y1, . . . , xnyn)(z1, . . . , zn)
= ((x1y1)z1, . . . , (xnyn)zn)
= (x1(y1z1), . . . , xn(ynzn))
= (x1, . . . , xn)(y1z1, . . . , ynzn)
= (x1, . . . , xn) [(y1, . . . , yn)(z1, . . . , zn)] .
• Identity is (e1, . . . , en), where ei is the identity of Gi.
• Inverse of (x1, . . . , xn) is (x−1
1 , . . . , x−1
n ).
16

Examples
1. Some new groups: C2×C2, C2×C2×C2, S4×D36, A5×A6×S297, . . . , Z×
Q × S13, . . ..
2. Consider C2 × C2. Elements are {(1, 1), (1, −1), (−1, 1), (−1, −1)}.
Calling these e, a, b, ab, mult table is
e a b ab
e e a b ab
a a e ab b
b b ab e a
ab ab b a e
G = C2 × C2 is abelian and x2 = e for all x ∈ G.
3. Similarly C2 × C2 × C2 has elements (±1, ±1, ±1), size 8, abelian,
x2 = e for all x.
Proposition 5.2 (a) Size of G1 × ∙ ∙ ∙ × Gn is |G1||G2| ∙ ∙ ∙ |Gn|.
(b) If all Gi are abelian so is G1 × ∙ ∙ ∙ × Gn.
(c) If x = (x1, . . . , xn) ∈ G1×∙ ∙ ∙×Gn, then order of x is the least common
multiple of o(x1), . . . , o(xn).
Proof (a) Clear.
(b) Suppose all Gi are abelian. Then
(x1, . . . , xn)(y1, . . . , yn) = (x1y1, . . . , xnyn)
= (y1x1, . . . , ynxn)
= (y1, . . . , yn)(x1, . . . , xn).
(c) Let ri = o(xi). Recall from M1P2 that xk
i = e iff ri|k. Let r =
lcm(r1, . . . , rn). Then
xr = (xr
1, . . . , xr
n)
= (e1, . . . , en) = e.
For 1 ≤ s < r, ri |s for some i. So xs
i = e. So
xs
= (. . . , xs
i , . . .) = (e1, . . . , en).
Hence r = o(x).
Examples
17

1. Since cyclic groups Cr are abelian, so are all direct products
Cr1 × Cr2 × ∙ ∙ ∙ × Crk
.
2. C4 × C2 and C2 × C2 × C2 are abelian of size 8. Are they isomorphic?
Claim: NO.
Proof Count the number of elements of order 2 :
In C4 × C2 these are (±1, ±1) except for (1, 1), so there are 3.
In C2 × C2 × C2, all the elements except e have order 2, so there
are 7.
So C4 × C2
∼= C2 × C2 × C2.
Proposition 5.3 If hcf(m, n) = 1, then Cm × Cn
∼= Cmn.
Proof Let Cm = α , Cn = β . So o(α) = m, o(β) = n. Consider
x = (α, β) ∈ Cm × Cn.
By 5.2(c), o(x) = lcm(m, n) = mn. Hence cyclic subgroup x of Cm × Cn
has size mn, so is whole of Cm × Cn. So Cm × Cn = x is cyclic and hence
Cm × Cn
∼= Cmn by 2.2.
Direct products are fundamental to the theory of abelian groups:
Theorem 5.4 Every finite abelian group is isomorphic to a direct product
of cyclic groups.
Won’t give a proof here. Reference: [Allenby, p. 254].
Examples
1. Abelian groups of size 6: by theorem 5.4, possibilities are C6, C3 × C2.
By 5.3, these are isomorphic, so there is only one abelian group of size
6 (up to isomorphism).
2. By 5.4, the abelian groups of size 8 are: C8, C4 × C2, C2 × C2 × C2.
Claim : No two of these are isomorphic.
Proof
Group C2 × C2 × C2 C4 × C2 C8
| {x | o(x) = 2} | 7 3 1
So up to isomorphism, there are 3 abelian groups of size 8.
18

6 Groups of small size
We’ll find all groups of size ≤ 7 (up to isomorphism). Useful results:
Proposition 6.1 If |G| = p, a prime, then G ∼= Cp.
Proof By corollary of Lagrange, G is cyclic. Hence G ∼= Cp by 2.2.
Proposition 6.2 If |G| is even, then G contains an element of order 2.
Proof Suppose |G| is even and G has no element of order 2. List the
elements of G as follows:
e, x1, x−1
1 , x2, x−1
2 , . . . , xk, x−1
k .
Note that xi = x−1
i since o(xi) = 2. Hence |G| = 2k + 1, a contradiction.
Groups of size 1, 2, 3, 5, 7
By 6.1, only such groups are C1, C2, C3, C5, C7.
Groups of size 4
Proposition 6.3 The only groups of size 4 are C4 and C2 × C2.
Proof Let |G| = 4. By Lagrange, every element of G has order 1, 2 or 4.
If there exists x ∈ G of order 4, then x is cyclic, so G ∼= C4. Now suppose
o(x) = 2 for all x = e, x ∈ G. So x2 = e for all x ∈ G.
Let e, x, y be 3 distinct elements of G. If xy = e then y = x−1 = x, a
contradiction; if xy = x then y = e, a contradiction; similarly xy = y. It
follows that
G = {e, x, y, xy} .
As above, yx = e, x, y hence yx = xy. So multiplication table of G is
e x y xy
e e x y xy
x x e xy y
y y xy e x
xy xy y x e
19

This is the same as the table for C2 × C2, so G ∼= C2 × C2.
Groups of size 6
We know the following groups of size 6: C6, D6, S3. Recall D6 is the
symmetry group of the equilateral triangle and has elements
e, ρ, ρ2
, σ, ρσ, ρ2
σ.
satisfying the following equations:
ρ3 = e,
σ2 = e
σρ = ρ2σ.
The whole multiplication table of D6 can be worked out using these equa-
tions. e.g.
σ ∙ (ρσ) = ρ2
σσ = ρ2
.
Proposition 6.4 Up to isomorphism, the only groups of size 6 are C6 and
D6.
Proof Let G be a group with |G| = 6. By Lagrange, every element of G
has order 1, 2, 3 or 6. If there exists x ∈ G of order 6, then G = x is cyclic
and therefore G ∼= C6 by 2.2. So assume G has no elements of order 6. Then
every x ∈ G, (x = e) has order 2 or 3. If all have order 2 then x2 = e for all
x ∈ G. So by Sheet 2 Q5, |G| is divisible by 4, a contradiction. We conclude
that there exists x ∈ G with o(x) = 3. Also by 6.2, there is an element y of
order 2.
Let H = x = e, x, x2 . Then y /∈ H so Hy = H and
G = H ∪ Hy = e, x, x2
, y, xy, x2
y .
What is yx? Well,
yx = e ⇒ y = x−1
yx = x ⇒ y = e
yx = x2 ⇒ y = x
yx = y ⇒ x = e



a contradiction.
If yx = xy, let’s consider the order of xy:
(xy)2
= xyxy = xxyy (as yx = xy) = x2
y2
= x2
.
20

Similarly
(xy)3
= x3
y3
= y = e.
So xy does not have order 2 or 3, a contradiction. Hence yx = xy. We
conclude that yx = x2y.
At this point we know the following:
• G = e, x, x2, y, xy, x2y ,
• x3 = e, x2 = e, yx = x2y.
In exactly the same way as for D6, can work out the whole multiplication
table for G using these equations. It will be the same as the table for D6
(with x, y instead of ρ, σ). So G ∼= D6.
Remark Note that |S3| = 6, and S3
∼= D6.
Summary
Proposition 6.5 Up to isomorphism, the groups of size ≤ 7 are
Size Groups
1 C1
2 C2
3 C3
4 C4, C2 × C2
5 C5
6 C6, D6
7 C7
Remarks on larger sizes
Size 8: here are the groups we know:
Abelian C8, C4 × C2, C2 × C2 × C2,
Non-abelian D8.
Any others? Yes, the quaternion group Q8:
Define matrices
A =
i 0
0 −i
, B =
0 1
−1 0
.
21

Check equations:
A4
= I, B4
= I, A2
= B2
, BA = A4
B.
Define
Q8 = {ArBs | r, s ∈ Z}
= {AmBn | 0 ≤ m ≤ 3, 0 ≤ n ≤ 1} .
Sheet 3 Q5: |Q8| = 8. Q8 is a subgroup of GL(2, C) and is not abelian and
Q8
∼= D8. Call Q8 the quaternion group. Sheet 3 Q7: The only non-abelian
groups of size 8 are D8 and Q8. Yet more info:
Size Groups
9 only abelian (Sh3 Q4)
10 C10, D10
11 C11
12 abelian, D12, A4 + one more
13 C13
14 C14, D14
15 C15
16 14 groups
7 Homomorphisms, normal subgroups and factor
groups
Homomorphisms are functions between groups which “preserve multiplica-
tion”.
Definition Let G, H be groups. A function φ : G → H is a homomorphism
if φ(xy) = φ(x)φ(y) for all x, y ∈ G.
Note that an isomorphism is a homomorphism which is a bijection.
Examples
1. G, H any groups. Define φ : G → H by
φ(x) = eH∀x ∈ G
Then φ is a homomorphism since φ(xy) = eH = eHeH = φ(x)φ(y).
22

2. Recall the signature function sgn : Sn → C2. By 4.1(a), sgn(xy) =
sgn(x)sgn(y), so sgn is a homomorphism.
3. Define φ : (R, +) → (C∗, ×) by
φ(x) = e2πix
∀x ∈ R.
Then φ(x + y) = e2πi(x+y) = e2πixe2πiy = φ(x)φ(y), so φ is a homo-
morphism.
4. Define φ : D2n → C2 (writing D2n = e, ρ, . . . , ρn−1, σ, ρσ, . . . , ρn−1σ )
by
φ(ρr
σs
) = (−1)s
.
(so φ sends rotations to +1 and reflections to −1). Then φ is a homo-
morphism since:
φ (ρrσs)(ρtσu) = φ(ρr±tσs+u)
= (−1)s+u = φ(ρrσs)φ(ρrσu).
Proposition 7.1 Let φ : G → H be a homomorphism
(a) φ(eG) = eH
(b) φ(x−1) = φ(x)−1 for all x ∈ G.
(c) o(φ(x)) divides o(x) for all x ∈ G.
Proof (a) Note that φ(eG) = φ(eGeG) = φ(eG)φ(eG). Multiply by φ(eG)−1
to get eH = φ(eG).
(b) By (a), eH = φ(eG) = φ(xx−1) = φ(x)φ(x−1). So φ(x−1) = φ(x)−1.
(c) Let r = o(x). Then
φ(x)r
= φ(x) ∙ ∙ ∙ φ(x) = φ(x ∙ ∙ ∙ x) = φ(xr
) = φ(eG) = eH.
Hence o(φ(x)) divides r.
Definition Let φ : G → H be homomorphism. The image of φ is
Imφ = φ(G) = {φ(x) | x ∈ G} ⊆ H.
Proposition 7.2 If φ : G → H is a homomorphism, then Imφ is a subgroup
of H.
23

Proof
(1) eH ∈ Imφ since eH = φ(eG).
(2) Let g, h ∈ Imφ. Then g = φ(x) and h = φ(y) for some x, y ∈ G, so
gh = φ(x)φ(y) = φ(xy) ∈ Imφ.
(3) Let g ∈ Imφ. Then g = φ(x) for some x ∈ G. So g−1 = φ(x)−1 =
φ(x−1) ∈ Imφ.
Hence Imφ is a subgroup of H.
Examples
1. Is there a homomorphism φ : S3 → C3? Yes, φ(x) = 1 for all x ∈ S3.
For this homomorphism, Imφ = {1}.
2. Is there a homomorphism φ : S3 → C3 such that Imφ = C3?
To answer this, suppose φ : S3 → C3 is a homomorphism. Consider
φ(1 2). By 7.1(c), φ(1 2) has order dividing o(1 2) = 2. As φ(1 2) ∈ C3,
this implies that φ(1 2) = 1. Similarly φ(1 3) = φ(2 3) = 1. Hence
φ(1 2 3) = φ ((1 3)(1 2)) = φ(1 3)φ(1 2) = 1
and similarly φ(1 3 2) = 1. We’ve shown that
φ(x) = 1∀x ∈ S3.
So there is no surjective homomorphism φ : S3 → C3.
Kernels
Definition Let φ : G → H be a homomorphism. Then kernel of φ is
Kerφ = {x ∈ G | φ(x) = eH} .
Examples
1. If φ : G → H is φ(x) = eH for all x ∈ G, then Kerφ = G.
2. For sgn : Sn → C2,
Ker(sgn) = {x ∈ Sn | sgn(x) = 1} = An, the alternating group.
24

3. If φ : (R, +) → (C∗, ×) is φ(x) = e2πix for all x ∈ R, then
Kerφ = x ∈ R | e2πix
= 1 = Z.
4. Let φ : D2n → C2 be given by φ(ρrσs) = (−1)s. Then Kerφ = ρ .
Proposition 7.3 If φ : G → H is a homomorphism, then Kerφ is a sub-
group of G.
Proof
(1) eG ∈ Kerφ as φ(eG) = eH by 7.1.
(2) x, y ∈ Kerφ then φ(x) = φ(y) = eH, so φ(xy) = φ(x)φ(y) = eH; i.e.
xy ∈ Kerφ.
(3) x ∈ Kerφ then φ(x) = eH, so φ(x)−1 = φ(x−1) = eH, so x−1 ∈ Kerφ.
In fact, Kerφ is a very special type of subgroup of G known as a normal
subgroup.
Normal subgroups
Definition Let G be a group, and N ⊆ G. We say N is a normal subgroup
of G if
(1) N is a subgroup of G,
(2) g−1Ng = N for all g ∈ G, where g−1Ng = g−1ng | n ∈ N .
If N is a normal subgroup of G, write N G.
Examples
1. G any group. Subgroup e = {e} G as g−1eg = e for all g ∈ G. Also
subgroup G itself is normal, i.e. G G, as g−1Gg = G for all g ∈ G.
Next lemma makes condition (2) a bit easier to check.
Lemma 7.4 Let N be a subgroup of G. Then N G if and only if g−1Ng ⊆
N for all g ∈ G.
25

Proof
⇒ Clear.
⇐ Suppose g−1Ng ⊆ N for all g ∈ G. Let g ∈ G. Then
g−1
Ng ⊆ N.
Using g−1 instead, we get (g−1)−1Ng−1 ⊆ N, hence
gNg−1
⊆ N.
Hence N ⊆ g−1Ng. Therefore g−1Ng = N.
Examples (1) We show that An Sn. Need to show that
g−1
Ang ⊆ An∀g ∈ Sn
(this will show An Sn by 7.4).
For x ∈ An, using 4.1 we have
sgn(g−1
xg) = sgn(g−1
)sgn(x)sgn(g) = sgn(g−1
) ∙ 1 ∙ sgn(g) = 1.
So g−1xg ∈ An for all x ∈ An. Hence
g−1
Ang ⊆ An.
So An Sn.
(2) Let G = S3, N = (1 2) = {e, (1 2)}. Is N G? Well,
(1 3)−1
(1 2)(1 3) = (1 3)(1 2)(1 3) = (2 3) /∈ N.
So (1 3)−1N(1 3) = N and N S3.
(3) If G is abelian, then all subgroups N of G are normal since for g ∈ G,
n ∈ N,
g−1
ng = g−1
gn = n,
and hence g−1Ng = N.
(4) Let D2n = e, ρ, . . . , ρn−1, σ, ρσ, . . . , ρn−1σ . Fix an integer r. Then
ρr
D2n.
Proof – sheet 4. (key: magic equation σρ = ρ−1σ, . . . , σρn = ρ−nσ).
26

Proposition 7.5 If φ : G → H is a homomorphism, then Kerφ G.
Proof Let K = Kerφ. By 7.3 K is a subgroup of G. Let g ∈ G, x ∈ K.
Then
φ(g−1
xg) = φ(g−1
)φ(x)φ(g) = φ(g)−1
eHφ(g) = eH.
So g−1xg ∈ Kerφ = K. This shows g−1Kg ⊆ K. So K G.
Examples
1. We know that sgn : Sn → C2 is a homomorphism, with kernel An. So
An Sn by 7.5.
2. Know φ : D2n → C2 defined by φ(ρrσs) = (−1)s is a homomorphism
with kernel ρ . So ρ D2n.
3. Here’s a different homomorphism α : D8 → C2 where
α(ρr
σs
) = (−1)r
.
This is a homomorphism, as
α((ρrσs)(ρtσu)) = α(ρr±tσs+u)
= (−1)r±t = (−1)r ∙ (−1)t
= α(ρrσs)α(ρtσu).
The kernel of α is
Kerα = {ρr
σs
| r even} = e, ρ2
, σ, ρ2
σ .
Hence
e, ρ2
, σ, ρ2
σ D8.
Factor groups
Let G be a group, N a subgroup of G. Recall that there are exactly |G|
|N|
different right cosets Nx (x ∈ G). Say
Nx1, Nx2, . . . , Nxr
where r = |G|
|N| . Aim is to make this set of right cosets into a group in a
natural way. Here is a “natural” definition of multiplication of these cosets:
(Nx)(Ny) = N(xy). (33)
27

Does this definition make sense? To make sense, we need:
Nx = Nx
Ny = Ny
⇒ Nxy = Nx y
for all x, y, x , y ∈ G. This property may or may not hold.
Example G = S3, N = (1 2) = {e, (1 2)}. The 3 right cosets of N in G
are
N = Ne, N(1 2 3), N(1 3 2).
Also
N = N(1 2)
N(1 2 3) = N(1 2)(1 2 3) = N(2 3)
N(1 3 2) = N(1 2)(1 3 2) = N(1 3).
According to (33),
(N(1 2 3)) (N(1 2 3)) = N(1 2 3)(1 2 3) = N(1 3 2).
But (33) also says that
(N(2 3)) (N(2 3)) = N(2 3)(2 3) = Ne.
So (33) makes no sense in this example.
How do we make (33) make sense? The condition is that N G. Key is
to prove the following:
Proposition 7.6 Let N G. Then for x1, x2, y1, y2 ∈ G
Nx1 = Nx2
Ny1 = Ny2
⇒ Nx1y1 = Nx2y2.
(Hence definition of multiplication of cosets in (33) makes sense when N
G.)
To prove this we need a definition and a lemma: for H a subgroup of G
and x ∈ G define the left coset
xH = {xh : h ∈ H}.
Lemma 7.7 Suppose N G. Then xH = Hx for all x ∈ G.
28

Proof Let h ∈ H. As H G, xHx−1 = H, and so xhx−1 = h ∈ H.
Then xh = h x ∈ Hx. This shows that xH ⊆ Hx. Similarly we see that
Hx ⊆ xH, hence xH = Hx.
Proof of Prop 7.6
Let N G. Suppose Nx1 = Nx2 and Ny1 = Ny2. Then
Nx1y1 = Nx2y1 as Nx1 = Nx2
= x2Ny1 by Prop 7.7
= x2Ny2 as Ny1 = Ny2
= Nx2y2 by Prop 7.7.
So we have established that when N G, the definition of multiplication
of cosets
(Nx)(Ny) = Nxy
for x, y ∈ G makes sense.
Theorem 7.8 Let N G. Define G/N to be the set of all right cosets Nx
(x ∈ G). Define multiplication on G/N by
(Nx)(Ny) = Nxy.
Then G/N is a group under this multiplication.
Proof
Closure obvious.
Associativity Using associativity in G
(NxNy)Nz = (Nxy)Nz
= N(xy)z
= Nx(yz)
= (Nx)(Nyz)
= Nx(NyNz).
Identity is Ne = N, since NxNe = Nxe = Nx and NeNx = Nex =
Nx.
Inverse of Nx is Nx−1, as NxNx−1 = Nxx−1 = Ne, the identity.
29

Definition The group G/N is called the factor group of G by N.
Note that
|G/N| =
|G|
|N|
.
Examples
1. An Sn. Since |Sn|
|An| = 2, the factor group Sn/An has 2 elements
An, An(1 2).
So Sn/An
∼= C2. Note: in the group Sn/An the identity is the coset
An and the non identity element An(1 2) has order 2 as
(An(1 2))2
= An(1 2)An(1 2) = An(1 2)(1 2) = An.
2. G any group. We know that G G. What is the factor group G/G?
Ans: G/G has 1 element, the identity coset G. So G/G ∼= C1.
Also e = {e} G. What is G/ e ? Coset e g = {g}, and multipli-
cation
( e g) ( e h) = e gh.
So G/ e ∼= G (isomorphism g → e g).
3. G = D12 = e, ρ, . . . , ρ5, σ, σρ, . . . , σρ5 where ρ6 = σ2 = e, σρ =
ρ−1σ.
(a) Know that ρ D12. Factor group D12/ ρ has 2 elements
ρ , ρ σ so D12/ ρ ∼= C2.
(b) Know also that ρ2 = e, ρ2, ρ4 D12. So D12/ ρ2 has 4
elements, so
D12/ ρ2 ∼= C4 or C2 × C2.
Which? Well, let N = ρ2 . The 4 elements of D12/N are
N, Nρ, Nσ, Nρσ.
We work out the order of each of these elements of D12/N:
(Nρ)2 = NρNρ = Nρ2
= N,
(Nσ)2 = NσNσ = Nσ2
= N,
(Nρσ)2 = N(ρσ)2
= N.
30

So all non-identity elements of D12/N have order 2, hence D12/ ρ ∼=
C2 × C2.
(c) Also ρ3 = e, ρ3 D12. Factor group D12 ρ3 has 6 elements
so is ∼= C6 or D6. Which? Let M = ρ3 . The 6 elements of
D12/M are
M, Mρ, Mρ2
, Mσ, Mρσ, Mρ2
σ.
Let x = Mρ and y = Mσ. Then
x3 = (Mρ)3 = MρMρMρ = Mρ3
= M,
y2 = (Mσ)2 = Mσ2
= M,
yx = MσMρ = Mσρ = Mρ−1σ = Mρ−1Mσ
= x−1y.
So D12/M = identity, x, x2, y, xy, x2y and x3 = y2 = identity,yx =
x−1y. So D12/ ρ3 ∼= D6.
Here’s a result tying all these topics together:
Theorem 7.9 (First Isomorphism Theorem) Let φ : G → H be a ho-
momorphism. Then
G/Kerφ ∼= Imφ.
Proof Let K = Kerφ. So G/K is the group consisting of the cosets
Kx (x ∈ G) with multiplication (Kx)(Ky) = Kxy. We want to define a
“natural” function G/K → Imφ. Obvious choice is the function Kx → φ(x)
for x ∈ G. To show this is a function, need to prove:
Claim 1. If Kx = Ky, then φ(x) = φ(y).
To prove this, suppose Kx = Ky. Then xy−1 ∈ K (as x ∈ Kx ⇒ x = ky
for some k ∈ K ⇒ xy−1 = k ∈ K ). Hence xy−1 ∈ K = Kerφ, so
φ(xy−1) = e
⇒ φ(x)φ(y−1) = e
⇒ φ(x)φ(y)−1 = e
⇒ φ(x) = φ(y).
By Claim 1, we can define a function α : G/K → Imφ by
α(Kx) = φ(x)
31

for all x ∈ G.
Claim 2. α is an isomorphism.
Here is a proof of this claim.
(1) α is surjective: for if φ(x) ∈ Imφ then φ(x) = α(Kx).
(2) α is injective:
α(Kx) = α(Ky)
⇒ φ(x) = φ(y)
⇒ φ(x)φ(y)−1 = e
⇒ φ(xy−1) = e,
so xy−1 ∈ Kerφ = K and so Kx = Ky.
(3) Finally
α((Kx)(Ky)) = α(Kxy)
= φ(xy)
= φ(x)φ(y)
= α(Kx)α(Ky).
Hence α is an isomorphism.
This completes the proof that G/K ∼= Imφ.
Corollary 7.10 If φ : G → H is a homomorphism, then
|G| = |Kerφ| ∙ |Imφ|.
One can think of this as the group theoretic version of the rank-nullity
theorem.
Examples
1. Homomorphism sgn : Sn → C2. By 7.9
Sn/Ker(sgn) ∼= Im(sgn),
so
Sn/An
∼= C2.
2. Homomorphism φ : (R, +) → (C∗, ×)
φ(x) = e2πix
.
32

Here
Kerφ = x ∈ R | e2πix = 1
= Z,
Imφ = e2πix | x ∈ R
= T the unit circle.
So R/Z ∼= T.
3. Is there a surjective homomorphism φ from S3 onto C3? Shown pre-
viously – No.
Here’s a better way to see this: suppose there exist such φ. Then
Imφ = C3, so by 7.9, S3/Kerφ ∼= C3. So Kerφ is a normal subgroup
of S3 of size 2. But S3 has no normal subgroups of size 2 (they are
(1 2) , (1 3) , (2 3) ).
Given a homomorphism φ : G → H, we know Kerφ G. Converse
question: Given a normal subgroup N G, does there exist a homomorphism
with kernel N? Answer is YES:
Proposition 7.11 Let G be a group and N G. Define H = G/N. Let
φ : G → H be defined by
φ(x) = Nx
for all x ∈ G. Then φ is a homomorphism and Kerφ = N.
Proof First, φ(xy) = Nxy = (Nx)(Ny) = φ(x)φ(y), so φ is a homomor-
phism. Also
x ∈ Kerφ ⇔ φ(x) = eH ⇔ Nx = N ⇔ x ∈ N.
Hence Kerφ = N.
Example From a previous example, we know ρ2 = e, ρ2, ρ4 D12. We
showed that D12 ρ2 ∼= C2 × C2. So by 7.11, the function φ(x) = ρ2 x
(x ∈ D12) is a homomorphism D12 → C2 × C2 which is surjective, with
kernel ρ2 .
Summary
There is a correspondence
{normal subgroups of G} ↔ {homomorphisms of G} .
33

For N G there is a homomorphism φ : G → G/N with Kerφ = N. For a
homomorphism φ, Kerφ is a normal subgroup of G.
Given G, to find all H such that there exist a surjective homomorphism
G → H:
(1) Find all normal subgroups of G.
(2) The possible H are the factor groups G/N for N G.
Example: G = S3.
(1) Normal subgroups of G are
e , G, A3 = (1 2 3)
(cyclic subgroups of size 2 (i j) are not normal).
(2) Factor groups:
S3/ e ∼= S3, S3/S3
∼= C1, S3/A3
∼= C2
8 Symmetry groups in 3 dimensions
These are defined similarly to symmetry groups in 2 dimensions, see chapter
2. An isometry of R3 is a bijection f : R3 → R3 such that d(x, y) =
d(f(x), f(y)) for all x, y ∈ R3.
Examples of isometries are: rotation about an axis, reflection in a plane,
translation.
As in 2.1, the set of all isometries of R3, under composition, forms a group
I(R3). For Π ⊆ R3, the symmetry group of Π is G(Π) = g ∈ I(R3) | g(Π) = Π .
There exist many interesting symmetry groups in R3. Some of the most in-
teresting are the symmetry groups of the Platonic solids: tetrahedron, cube,
octahedron, icosahedron, dodecahedron.
Example: The regular tetrahedron
Let Π be regular tetrahedron in R3, and let G = G(Π).
• Rotations in G: Let R be the set of rotations in G. Some elements of
R:
34

(1) e,
(2) rotations of order 3 fixing one corner: these are
ρ1, ρ2
1, ρ2, ρ2
2, ρ3, ρ2
3, ρ4, ρ2
4
(where ρi fixes corner i),
(3) rotations of order 2 about an axis joining the mid-points of op-
posite sides
ρ12,34, ρ13,24, ρ14,23.
So |R| ≥ 12. Also |R| ≤ 12: can rotate to get any face i on bottom (4
choices). If i is on the bottom, only 3 possible configurations. Hence
|R| ≤ 4 ∙ 3 = 12. Hence |R| = 12.
Claim 1: R ∼= A4.
To see this, observe that each rotation r ∈ R gives a permutation of
the corners 1, 2, 3, 4, call it πr:
e → πe = identity permutation
ρi, ρ2
i → all 8 3-cycles in S4 (1 2 3), (1 3 2), . . .
ρ12,34 → (1 2)(3 4)
ρ13,24 → (1 3)(2 4)
ρ14,23 → (1 4)(2 3).
Notice that {πr | r ∈ R} consists of all the 12 even permutations in
S4, i.e. A4. The map r → πr is an isomorphism R → A4. So R ∼= A4.
Claim 2: The symmetry group G is S4.
Obviously G contains a reflection σ with corresponding permutation
πσ = (1 2). So G contains
R ∪ Rσ.
So |G| ≥ |R| + |Rσ| = 24. On the other hand, each g ∈ G gives a
unique permutation πg ∈ S4, so |G| ≤ |S4| = 24. So |G| = 24 and the
map g → πg is an isomorphism G → S4.
9 Counting using groups
Consider the following problem. Colour edges of an equilateral triangle with
2 colours R, B. How many distinguishable colourings are there?
Answer: There are 8 colourings altogether:
35

(1) all the edges red – RRR,
(2) all the edges blue – BBB,
(3) two reds and a blue – RRB,RBR,BRR,
(4) two blues and a red – BBR,BRB,RBB.
Clearly there are 4 distinguishable colourings. Point: Two colourings are
not distinguishable iff there exists a symmetry of the triangle sending one
to the other.
To bring groups into the picture: call C the set of all 8 colorings. So
C = {RRR, . . . , RBB} .
Let G be the symmetry group of the equilateral triangle, D6 = e, ρ, ρ2, σ, ρσ, ρ2σ .
Each element of D6 gives a permutation of C, e.g. ρ gives the permutation
(RRR) (BBB) (RRB RBR BRR) (BBR BRB RBB).
Divide the set C into subsets called orbits of G: two colourings c, d are
in the same orbit if there exists g ∈ D6 sending c to d. The orbits are the
sets (1) - (4) above. The number of distinguishable colourings is equal to
the number of orbits of G.
General situation
Suppose we have a set S and a group G consisting of some permutations
of S (e.g. S = C, G = D6 above). Partition S into orbits of G, by saying
that two elements s, t ∈ S are in the same orbit iff there exists a g ∈ G such
that g(s) = t. How many orbits are there?
Lemma 9.1 (Burnside’s Counting Lemma) For g ∈ G, define
fix(g) = number of elements of S fixed by g
= |{s ∈ S | g(s) = s}| .
Then
number of orbits of G =
1
|G|
g∈G
fix(g).
I won’t give a proof. Look it up in the recommended book by Fraleigh
if you are interested.
Examples
36

(1) C = set of 8 colourings of the equilateral triangle. G = D6. Here are
the values of fix(g):
g e ρ ρ2 σ ρσ ρ2σ
fix(g) 8 2 2 4 4 4
By 9.1, number of orbits is 1
6 (8 + 2 + 2 + 4 + 4 + 4) = 4.
(2) 6 beads coloured R, R, W, W, Y, Y are strung on a necklace. How
many distinguishable necklaces are there?
Each necklace is a colouring of a regular hexagon. Two colourings are
indistinguishable if there is a rotation or reflection sending one to the
other (a reflection is achieved by turning the hexagon upside down).
Let D be the set of colourings of the hexagon and G = D12.
g e ρ ρ2 ρ3 ρ4 ρ5
fix(g) 6
2 × 4
2 0 0 6 0 0
g σ ρσ ρ2σ ρ3σ ρ4σ ρ5σ
fix(g) 6 6 6 6 6 6
So by 9.1
number of orbits =
1
12
(90 + 42) = 11.
So the number of distinguishable necklaces is 11.
(3) Make a tetrahedral die by putting 1, 2, 3, 4 on the faces. How many
distinguishable dice are there?
Each die is a colouring (colours 1, 2, 3, 4) of a regular tetrahedron. Two
such colourings are indistinguishable if there exists a rotation of the
tetrahedron sending one to the other. Let E be the set of colourings,
and G = rotation group of tetrahedron (so |G| = 12, G ∼= A4 by
Chapter 8). Here for g ∈ G
fix(g) =
24 if g = e,
0 if g = e.
So by 9.1, number of orbits is 1
12 (24) = 2. So there are 2 distinguishable
tetrahedral dice.
37

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