Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Outline
1
Factor group definition and examples
2
G/Z (G) cyclic ⇒ G abelian
3
Cauchy’s Theorem for abelian groups
College of St. Benedict/St. John’s University
Department of Mathematics
4
Direct product definition
Math 331
5
Direct products of groups are groups
6
Classification of groups of order p2
Chapter 9: Factor Groups
Bret Benesh
Bret Benesh
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Definition
Let N be a normal subgroup of a group G. The factor group (or
quotient group) G/N is the set of all right cosets of N in G
where the multiplication is defined to be (Nx)(Ny ) = Nxy for all
x, y ∈ G.
Bret Benesh
Chapter 9: Factor Groups
Chapter 9: Factor Groups
Example
Let G = D4 =
{e, (1, 2, 3, 4), (1, 3)(2, 4), (1, 4, 3, 2), (1, 2)(3, 4), (1, 4)(2, 3),
(2, 4), (1, 3)} and N = {e, (1, 3)(2, 4)}. Then the elements of
G/N = D4 /N are:
1
N = Ne = N(1, 3)(2, 4) = {e, (1, 3)(2, 4)}
2
N(1, 2, 3, 4) = N(1, 4, 3, 2) = {(1, 2, 3, 4), (1, 4, 3, 2)}
3
N(1, 2)(3, 4) = N(1, 4)(2, 3) = {(1, 2)(3, 4), (1, 4)(2, 3)}
4
N(1, 3) = N(2, 4) = {(1, 3), (2, 4)}
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Why does this form a group?
A better first question is “Why is this multiplication
well-defined?"
Example
We know that
1
2
=
2
4
and
3
5
=
15
25 .
Note that
2 15
1 3
+ = +
.
2 5
4 25
Example
We say that fraction addition is well-defined because
a
b
+
c
d
=
e
f
+
g
h
whenever ba = ef and dc = gh . Similarly, fraction multiplication is
well-defined—it does not matter which “name" you use to do
the computations.
Example (Non-example)
Define ba z dc =
4
1 3
2z5 = 7.
17
2 15
4 z 25 = 29
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
a+c
b+d .
6=
This operation is not well-defined because
4
7.
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
For the G = D4 and N = {e, (1, 3)(2, 4)} example, we should
be concerned that, since N(1, 3) = N(2, 4), say, it could be that
N(1, 2, 3, 4)N(1, 3) 6= N(1, 2, 3, 4)N(2, 4).
We will now show that they end up being equal (as long as N is
normal).
Theorem
The multiplication for factor groups is well-defined.
Proof.
Let N be a normal subgroup of a group G, and a, a0 , b, b0 ∈ G
such that Na = Na0 and Nb = Nb0 . Then a0 = na and b0 = mb
for some n, m ∈ N. Also, Na = aN, so there exists an m0 ∈ N
with am = m0 a. Finally,
(Na0 )(Nb0 ) = Na0 b0 = N(na)b0 = (Na)b0 = Na(mb)
= N(am)b = N(m0 a)b = N(ab) = (Na)(Nb).
Bret Benesh
Chapter 9: Factor Groups
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Example
Example (Non-example)
Let G = S3 , H = {e, (1, 2, 3), (1, 3, 2)}. There are two right
cosets, H = He and H(1, 2) = {(1, 2), (2, 3), (1, 3) }. There are
two left cosets, H = eH and (1, 2)H = {(1, 2), (1, 3), (2, 3) }.
Let G = S3 , H = {e, (2, 3)}. Note that H(1, 2) = {(1, 2), (1, 2, 3)
} and (1, 2)H = {(1, 2), (1, 3, 2) }. Then
(H(1, 2))Hx
(H(1, 2))Hx
= H((1, 2)Hx)
= H((1, 2)Hx)
= H((1, 2){e, (2, 3)})x
= H((1, 2){e, (1, 2, 3), (1, 3, 2)})x
= H({(1, 2), (1, 3, 2)})x
= H({(1, 2), (1, 3), (2, 3)})x
6= H({(1, 2), (1, 2, 3)})x
= H({(1, 2), (2, 3), (1, 3)})x
= H({e, (2, 3)}(1, 2))x
= H({e, (1, 2, 3), (1, 3, 2)}(1, 2))x
= H(H(1, 2)x)
= H(H(1, 2)x)
= H(1, 2)x
= H(1, 2)x
Bret Benesh
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Theorem
The factor group G/N is indeed a group.
Proof.
We will show that G/N satisfies the four group axioms. Let
a, b, c ∈ G.
1
2
(Na)(N) = (Na)(Ne) = N(ae) = Na = N(ea) =
(Ne)(Na) = (N)(Na). So the coset N is the identity.
(Na)(Na−1 ) = N(aa−1 ) = N = N(a−1 a) = (Na−1 )(Na). So
Na−1 is the inverse of Na for all a ∈ G.
3
(NaNb)Nc = (Nab)Nc = N(ab)c = Na(bc) = Na(Nbc) =
Na(NbNc). So G/N is associative.
4
NaNb = N(ab) ∈ G/N. So G/N is closed.
Bret Benesh
Chapter 9: Factor Groups
Chapter 9: Factor Groups
In your teams, find the elements and multiplication table for
C6 /C2 andA4 /V4 , where
A4 = {e, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3), (1, 2, 3), (1, 3, 2),
(1, 2, 4), (1, 4, 2), (1, 3, 4), (1, 4, 3), (2, 3, 4), (2, 4, 3)}
and
V4 = {e, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)}
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Multiplication Table for C6 /C2
Multiplication Table for A4 /V4
C2 ∼
= {0, 3} ⊆ C6
+
C2
C2 + 1
C2 + 2
C2
C2
C2 + 1
C2 + 2
C2 + 1
C2 + 1
C2 + 2
C2
C2 + 2
C2 + 2
C2
C2 + 1
V4 = {e, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)}
V4 (1, 2, 3) = {(1, 2, 3), (1, 3, 4), (2, 4, 3), (1, 4, 2)}
V4 (1, 3, 2) = {(1, 3, 2), (2, 3, 4), (1, 2, 4), (1, 4, 3)}
·
V4
V4 (1, 2, 3)
V4 (1, 3, 2)
Bret Benesh
V4
V4
V4 (1, 2, 3)
V4 (1, 3, 2)
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
V4 (1, 2, 3)
V4 (1, 2, 3)
V4 (1, 3, 2)
V4
Bret Benesh
V4 (1, 3, 2)
V4 (1, 3, 2)
V4
V4 (1, 2, 3)
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Theorem
If G/Z (G) is cyclic, then G is abelian.
Proof.
Suppose that G/Z (G) = hZ (G)ai for some a ∈ G with
o(Z (G)a) = n. Then
G = Z (G) ∪ Z (G)a ∪ Z (G)a2 ∪ · · · ∪ Z (G)an−1
and for all g, h ∈ G , g = z1 ai and h = z2 ak for some
z1 , z2 ∈ Z (G) and 0 ≤ i, k ≤ n − 1.
Proof (continued).
gh = (z1 ai )(z2 ak )
= z1 (ai z2 )ak
= z1 (z2 ai )ak
= z1 z2 (ai ak )
= z1 z2 (ak ai )
= z1 (z2 ak )ai
= (z2 ak )(z1 ai )
= hg
Bret Benesh
Chapter 9: Factor Groups
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Theorem
Suppose that G is abelian, and p is a prime that divides |G|.
Then G has an element of order p.
Proof.
We will induct on G. If |G| = 1, this result is vacuously true. So
suppose |G| > 1, and assume that the result is true of groups
of order less than |G| that fulfill the hypotheses.
Let e 6= g ∈ G, and suppose q is a prime number that divides
o(g). Then o(g o(g)/q ) = q. So without loss of generality, we
may say o(g) = q.
Proof (continued).
Since G is abelian, hgi is a normal subgroup; consider
H = G/hgi. The factor group H is abelian, and p divides
|H| = |G|
q . So H has an element (hgia) (for some a ∈ G) of
order p by the induction hypothesis.
Suppose o(a) = n in G. Then (hgia)n = hgian = hgie = hgi.
Since hhgiai is a cyclic group of order p, this implies that p
divides n (by a previous result about cyclic groups), so n = pm
for some integer m.
Then o(am ) = p.
If q = p, we are done. So assume q 6= p.
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Definition
If G and H are groups, the direct product G × H is the set
{(g, h) | g ∈ G, h ∈ H}.
Theorem
For all groups G and H, G × H is a group where
(g1 , h1 )(g2 , h2 ) = (g1 g2 , h1 h2 ) for all gi ∈ G, hi ∈ H.
Example
Proof.
It is easy to check that the identity is (eG , eH ) and
(g, h)−1 = (g −1 , h−1 ) for all g ∈ G, h ∈ H. It is clearly closed by
the definition of its multiplication, so it remains to check
associativity. Let g1 , g2 , g3 ∈ G and h1 , h2 , h3 ∈ H. Then
The Klein-4 group is C2 × C2 .
((g1 , h1 )(g2 , h2 ))(g3 , h3 ) = (g1 g2 , h1 h2 )(g3 , h3 ) =
((g1 g2 )g3 , (h1 h2 )h3 ) = (g1 (g2 g3 ), h1 (h2 h3 )) =
(g1 , h1 )(g2 g3 , h2 h3 ) = (g1 , h1 )((g2 , h2 )(g3 , h3 )).
Bret Benesh
Chapter 9: Factor Groups
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Lemma (Lemma 2)
Lemma (Lemma 1)
If N and M are normal subgroups of a group G such that
N ∩ M = e, then nm = mn for all n ∈ N, m ∈ M.
Proof.
Let n ∈ N and m ∈ M, and consider the element
[n, m] = n−1 m−1 nm. Then
(m−1 )n m = n−1 m−1 nm = n−1 (n)m
If N and M are normal subgroups of G with N ∩ M = e, then the
subgroup NM is isomorphic to N × M.
Proof.
We will define a map φ : NM → N × M by φ(nm) = (n, m). This
map is clearly onto, and can easily be seen to be one-to-one.
It remains to show that φ is operation-preserving, so consider
n1 m1 , n2 m2 ∈ NM where the ni ∈ N and mi ∈ M for i = 1, 2.
Then
So [n, m] ∈ N ∩ M = e, and n−1 m−1 nm = e for all n ∈ N,
m ∈ M. So nm = mn.
Bret Benesh
φ((n1 m1 )(n2 m2 )) = φ(n1 m1 n2 m2 )
Chapter 9: Factor Groups
Bret Benesh
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Theorem
Proof (continued).
φ((n1 m1 )(n2 m2 )) = φ(n1 m1 n2 m2 )
Let G be a group. If |G| = p2 for some prime p, then G ∼
= Cp2 or
G∼
= Cp × Cp .
Proof.
Next, we will use the fact that m1 n2 = n2 m1 from Lemma 1.
φ(n1 m1 n2 m2 ) = φ(n1 n2 m1 m2 )
= (n1 n2 , m1 m2 )
= (n1 , m1 )(n2 , m2 )
= φ(n1 m1 )φ(n2 m2 )
Bret Benesh
Chapter 9: Factor Groups
If G has an element of order p2 , then G ∼
= Cp2 , and we are
done.
So assume G has no element of order p2 . By Lagrange’s
Theorem, every element of G must have either order 1 (only for
the identity) or order p. So let e 6= g ∈ G. Then o(g) = p. Let
A = hgi.
Let h be an element of G such that h ∈
/ hgi, and let B = hhi,
and note that A ∩ B = {e}.
Bret Benesh
Chapter 9: Factor Groups
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Factor group definition and examples
G/Z (G) cyclic ⇒ G abelian
Cauchy’s Theorem for abelian groups
Direct product definition
Direct products of groups are groups
Classification of groups of order p2
Proof (continued).
Proof (continued).
Let X = {g i hj | 0 ≤ i, j ≤ p − 1}. Then X has p2 elements, so
X = G.
Ax
= A for
We will prove that A is normal in G, so we will prove
all x ∈ G. But x ∈ G = X , so x = g i hj for some 0 ≤ i, j ≤ p − 1.
i j
It is enough to show that g x = g g h ∈ A.
Note that g g = g ∈ A, so it is really sufficient to show that
g h ∈ A. Suppose that g h ∈
/ A. Then o(g h ) = p, g ∈
/ hg h i, and G
is a union of cosets of the form hg h ig k for 0 ≤ k ≤ p − 1.
Bret Benesh
Chapter 9: Factor Groups
h k
Since h−1 ∈ G = ∪p−1
k =0 hg ig ,
h−1 = (g h )m g k = (h−1 g m h)g k
for some integer m.
Then h−1 = h−1 g m hg k , so e = g m hg k and g −m−k = h. This
means that h ∈ hgi, which is a contradiction. So g h ∈ A = hgi,
and A is normal.
The subgroup B = hhi is normal by a similar argument, and
A ∩ B = e. By Lemma 2, we conclude that
G∼
= Cp × Cp .
=A×B ∼
Bret Benesh
Chapter 9: Factor Groups