CHAPTER 9 Normal Subgroups and Factor Groups Normal Subgroups If H G, we have seen situations where aH 6= Ha 8 a 2 G. Definition (Normal Subgroup). A subgroup H of a group G is a normal subgroup of G if aH = Ha 8 a 2 G. We denote this by H C G. Note. This means that if H C G, given a 2 G and h 2 H , 9 h 0 ,h 00 2 H 3 -- ah = h 0 a and ah 00 = ha. and conversely. It does not mean ah = ha for all h 2 H . Recall (Part 8 of Lemma on Properties of Cosets). aH = Ha () H = aHa -1 . Theorem (9.1 — Normal Subgroup Test). If H G, H C G () xHx -1 ✓ H for all x 2 G. Proof. (= )) H C G = ) 8 x 2 G and 8 h 2 H , 9 h 0 2 H 3 -- xh = h 0 x = ) xhx -1 = h 0 2 H . Thus xHx -1 ✓ H . (( =) Suppose xHx -1 ✓ H 8 x 2 G. Let x = a. Then aHa -1 ✓ H = ) aH ✓ Ha. Now let x = a -1 . Then a -1 H (a -1 ) -1 = a -1 Ha ✓ H = ) Ha ✓ aH . By mutual inclusion, Ha = aH = ) H C G. ⇤ 117
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CHAPTER 9
Normal Subgroups and Factor Groups
Normal Subgroups
If H G, we have seen situations where aH 6= Ha 8 a 2 G.Definition (Normal Subgroup).
A subgroup H of a group G is a normal subgroup of G if aH = Ha 8 a 2 G.We denote this by H C G.
Note. This means that if H C G, given a 2 G and h 2 H, 9 h0, h00 2 H3�� ah = h0a and ah00 = ha. and conversely. It does not mean ah = ha for allh 2 H.
Recall (Part 8 of Lemma on Properties of Cosets).
aH = Ha () H = aHa�1.
Theorem (9.1 — Normal Subgroup Test).
If H G, H C G () xHx�1 ✓ H for all x 2 G.
Proof.
(=)) H C G =) 8 x 2 G and 8 h 2 H, 9 h0 2 H 3�� xh = h0x =)xhx�1 = h0 2 H. Thus xHx�1 ✓ H.
((=) Suppose xHx�1 ✓ H 8 x 2 G. Let x = a. Then aHa�1 ✓ H =)aH ✓ Ha. Now let x = a�1. Then a�1H(a�1)�1 = a�1Ha ✓ H =) Ha ✓aH.
By mutual inclusion, Ha = aH =) H C G. ⇤
117
118 9. NORMAL SUBGROUPS AND FACTOR GROUPS
Example.
(1) Every subgroup of an Abelian group is normal since ah = ha for all a 2 Gand for all h 2 H.
(2) The center Z(G) of a group is always normal since ah = ha for all a 2 Gand for all h 2 Z(G).
Theorem (4). If H G and [G : H] = 2, then H C G.
Proof.
If a 2 H, then H = aH = Ha. If a 62 H, aH is a left coset distinct from H andHa is a right coset distinct from H. Since [G : H] = 2, G = H[aH = H[Haand H \ aH = ; = H \Ha =) aH = Ha. Thus H C G. ⇤
Example. An C Sn since [Sn : An] = 2.
Note, for example, that for (1 2) 2 Sn and (1 2 3) 2 An,
(1 2)(1 2 3) 6= (1 2 3)(1 2),
but(1 2)(1 2 3) = (1 3 2)(1 2)
and (1 3 2) 2 An.
Example. hR360/ni C Dn since [Dn : R360/n] = 2.
Example. SL(2, R) C GL(2, R).
Proof.
Let x 2 GL(2, R). Recall det(x�1) =1
det(x)= (det(x))�1. Then, for all
h 2 SL(2, R),
det(xhx�1) = (det(x))(det(h))(det(x))�1 =
(det(x))(det(x))�1(det(h)) = 1 · 1 = 1,
so xhx�1 2 SL(2, R) =) x SL(2, R)x�1 ✓ SL(2, R). Thus SL(2, R) C GL(2, R).⇤
9. NORMAL SUBGROUPS AND FACTOR GROUPS 119
Example. Consider A4, with group table from page 111 shown below:
Let H = {↵1,↵2,↵3,↵4} A4, A = {↵5,↵6,↵7,↵8} ✓ A4,
and B = {↵9,↵10,↵11,↵12}. 8 a 2 A, a�1 2 B, and 8 b 2 B, b�1 2 A.
Let x 2 A4:
Case 1: x 2 H. Then xH ✓ H. Since x�1 2 H, xHx�1 ✓ H.
Case 2: x 2 A. Then xH ✓ A =) xHx�1 ✓ H.
Case 3: x 2 B. Then xH ✓ B =) xHx�1 ✓ H.
Thus, 8 x 2 A4, xHx�1 ✓ H =) H C A4 by Theorem 9.1.
Now let K = {↵1,↵5,↵9} A4. Now ↵5 2 K, but
↵2↵5↵�12 = ↵2↵5↵2 = ↵2↵8 = ↵7 62 K,
so K 6C A4.
120 9. NORMAL SUBGROUPS AND FACTOR GROUPS
Factor Groups
Theorem (9.2 — Factor Groups). Let G be a group and H C G. Theset G/H = {aH|a 2 G} is a group under the operation (aH)(bH) = abH.This group is called the factor group or quotient group of G by H.
Proof.
We first show the operation is well-defined. [Our product is determined by thecoset representatives chosen, but is the product uniquely determined by thecosets themselves?]
Suppose aH = a0H and bH = b0H. Then a0 = ah1 and b0 = bh2 for someh2, h2 2 H =) a0b0H = ah1bh2H = ah1bH = ah1Hb = aHb = abH byassociativity in G, the Lemma on cosets (page 145), and the fact that H C G.Thus the operation is well-defined.
Associativity in G/H follows directly from associativity in G:
In general, for n > 0, nZ = {0,±n,±2n,±3n, . . . }, and Z/nZ ⇡ Zn.
122 9. NORMAL SUBGROUPS AND FACTOR GROUPS
Example. Consider the multiplication table for A4 below, where i repre-sents the permutation ↵i on page 117 of these notes.
Let H = {1, 2, 3, 4}. The 3 cosets of H are H, 5H = {5, 6, 7, 8}, and 9H ={9, 10, 11, 12}. Notice how the above table is divided into coset blocks. SinceH C A4, when we replace the various boxes by their coset names, we get theCayley table below for A4/H.
The factor group collapses all the elements of a coset to a single group elementof A4/H.
When H C G, one can always arrange a Cayley table so this happens. WhenH 6C G, one cannot.
9. NORMAL SUBGROUPS AND FACTOR GROUPS 123
Example. Is U(30)/U5(30) isomorphic to Z2 � Z2 (the Klein 4-group) orZ4?
(3H)2 = 9H 6= H, do 3H has order at least 4, ruling out Z2 � Z2 � Z2.
(3H)4 = 17H 6= H, so |3H| 6= 4 =) |3H| = 8.
Thus G/H ⇡ Z8. ⇤
Applications of Factor Groups
Why are factor groups important? When G is finite and H 6= {e}, G/H issmaller than G, yet simulates G in many ways. One can often deduce propertiesof G from G/H.
Theorem (9.3 — G/Z Theorem). Let G be a group and let Z(G) be thecenter of G. If G/Z(G) is cyclic, then G is Abelian.
Proof.
Let gZ(G) be a generator of G/Z(G), and let a, b 2 G. Then 9 i, j 2 Z 3��aZ(G) = (gZ(G))i = giZ(G) and
bZ(G) = (gZ(G))j = gjZ(G). Thus
a = gix and b = gjy for some x, y 2 Z(G). Then
ab = (gix)(gjy) = gi(xgj)y = gi(gjx)y = (gigj)(xy) =
(gjgi)(yx) = (gjy)(gix) = ba.
Thus G is Abelian. ⇤
9. NORMAL SUBGROUPS AND FACTOR GROUPS 125
Note.
(1) From the proof above, we have a stronger e↵ect:
Theorem (9.30). Let G be a group and H Z(G). If G/H is cyclic,then G is Abelian.
(2) Contrapositive:
Theorem (9.300). If G is non-Abelian, then G/Z(G) is not cyclic.
Example. Consider a non-Abelian group of order pq, where p and q areprimes. Then, since G/Z(G) is not cyclic, |Z(G)| 6= p and |Z(G)| 6= q, so|Z(G)| = 1 and Z(G) = {e}.
(3) If G/Z(G) is cyclic, it must be trivial (only the identity).
Theorem (9.4 — G/Z(G) ⇡ Inn(G)). For any group G, G/Z(G) isisomorphic to Inn(G).
Proof.
Consider T : G/Z(G) ! INN(G) defined by T (gZ(G)) = �g.
[To show T is a well-defined function.] Suppose gZ(G) = hZ(G) =) h�1g 2Z(G). Then, 8 x 2 G,
Thus �g = �h. Reversing the above argument shows T is 1–1. T is clearlyonto. For operation preservation, suppose gZ(G), hZ(G) 2 G/Z(G). Then
T (gZ(G) · hZ(G)) = T (ghZ(G)) = �gh = �g�h = T (gZ(G)) · T (hZ(G)).
Thus T is an isomorphism. ⇤
126 9. NORMAL SUBGROUPS AND FACTOR GROUPS
Problem (Page 204 # 60). Find |Inn(Dn)|.Solution. By Example 14, page 67, Z(Dn) = {R0, R180} for n even and
Z(Dn) = {R0} for n odd. Thus, for n odd, Dn/Z(Dn) = Dn ⇡ Inn(Dn) byTheorem 9.4, |Inn(Dn)| = |Dn| = 2n for n odd.
Now suppose n is even. Then
|Dn/Z(Dn)| = [Dn : Z(Dn)] =|Dn|
|Z(Dn)|=
2n
2= n.
Then, since Dn/Z(Dn) ⇡ Inn(Dn) by Theorem 9.4,
|Inn(Dn)| = |Dn/Z(Dn)| = n.
Further, n = 2p, p a prime. Then, by Theorem 7.3, Inn(Dn) ⇡ Zn orInn(Dn) ⇡ Dp. If Inn(Dn) were cyclic, Dn/Z(Dn) would also be by Theo-rem 9.4 =) Dn is Abelian by Theorem 9.3, an impossibility.
Thus Inn(D2p) ⇡ Dp. ⇤
9. NORMAL SUBGROUPS AND FACTOR GROUPS 127
Theorem (9.5 — Cauchy’s Theorem for Abelian groups). Let G be a finiteAbelian group and let p be a prime such that p
��|G|. Then G has an elementof order p.
Proof.
Clearly, the theorem is true if |G| = 2. We use the Second Principle of Inductionon |G|. Assume the staement is true for all Abelian groups with order less than|G|. [To show, based on the induction assumption, that the statement holdsfor G also.]
Now G must have elements of prime order: if |x| = m and m = qn, where q isprime, then |xn| = q. Let x be an element of prime order q. If q = p, we arefinished, so assume q 6= p.
Since every subgroup of an Abelian group is normal, we may construct G =
G/hxi. Then G is Abelian and p��|G|, since |G| =
|G|q
. By induction, then, G
has an element – call it yhxi – of order p.
For the conclusion of the proof we use the following Lemma: ⇤
Lemma (Page 204 # 67). Suppose H C G, G finite. If G/H has anelement of order n, G has an element of order n.
Proof.
Suppose |gH| = n. Suppose |g| = m. Then (gH)m = gmH = eH = H, so byCorollary 2 to Theorem 4.1, n|m. [We just proved Page 202 # 37.] Then
9 t 2 Z 3�� m = |g| = nt = |gH|tso, by Theorem 4.2,
|gt| =m
gcd(m, t)=
m
t= n.
⇤
Example. Consider, for k 2 Z, hki C Z. 1+ hki 2 Z/hki with |1+ hki| =k, but all elements of Z have infinite order, so the assumption that G must befinite in the Lemma is necessary.
128 9. NORMAL SUBGROUPS AND FACTOR GROUPS
Internal Direct Products
Our object here is to break a group into a product of smaller groups.
Definition. G is the internal direct product of H and K and we writeG = H ⇥K if H C G, K C G, G = HK, and H \K = {e}.
Note.
(1) For an internal direct product, H and K must be di↵erent normal subgroupsof the same group.
(2)For external direct products, H and K can be any groups.
(3) One forms an internal direct product by starting with G, and then findingtwo normal subgroups H and K within G such that G is isomorphic to theexternal direct product of H and K.
9. NORMAL SUBGROUPS AND FACTOR GROUPS 129
Example. Consider Z35 where, as we recall, the group operation is addition.
h5i C Z35 and h7i C Z35 since Z35 is Abelian.
Since gcd(5, 7) = 1, 9 s, t 2 Z 3�� 1 = 5s + 7t. In fact, 1 = (�4)5 + (3)7.
Thus, for m 2 Z35, m = (�4m)5 + (3m)7 2 h5i + h7i. So Z35 = h5i + h7i.Also, h5i \ h7i = {0}, so Z35 = h5i ⇥ h7i.We also know h5i ⇡ Z7 and h7i ⇡ Z5, so that
Let H1, H2, . . . , Hn be a finite collection of normal subgroups of G. We saythat G is the internal direct product of H1, H2, . . . , Hn and write
G = H1 ⇥H2 ⇥ · · ·⇥Hn if
(1) G = H1H2 · · ·Hn = {h1h2 · · ·hn|hi 2 Hi},
(2) (H1H2 · · ·Hi) \Hi+1 = {e} for i = 1, 2, . . . , n� 1.
130 9. NORMAL SUBGROUPS AND FACTOR GROUPS
Theorem (9.6 — H1H2 · · ·Hn ⇡ H1 �H2 � · · ·�Hn). If a group G isthe internal direct product of a finite number of subgroups H1, H2, . . . , Hn,then G = H1 �H2 � · · ·�Hn, i.e.,
H1 ⇥H2 ⇥ · · ·⇥Hn ⇡ H1 �H2 � · · ·�Hn.
Proof.
[To show h’s from di↵erent Hi’s commute.] Let hi 2 Hi and hj 2 Hj withi 6= j. Then, since Hi C G and Hj C G,
(hihjh�1i )h�1
j 2 Hjh�1j = Hj and hi(hjh
�1i h�1
j ) 2 hiHi = Hi.
Thus hihjh�1i h�1
j 2 Hi \Hj = {e} by Page 200 # 5. [WLOG, suppose i < j.Then, if h 2 Hi\Hj, h 2 H1H2 · · ·Hi · · ·Hj�1\Hj = {e} from the definitionof internal direct product.] Thus hihj = hjhi.
[To show each element of G can be expressed uniquely in the form h1h2 · · ·hn
where hi 2 Hi.] From the definition of internal direct product, there existh1 2 H1, . . . , hn 2 Hn such that g = h1h2 · · ·hn for g 2 G. Suppose alsog = h01h
02 · · ·h0n where h0i 2 Hi. Using the commutative property shown above,
we can solve(?) h1h2 · · ·hn = h01h
02 · · ·h0n
to geth0nh
�1n = (h01)
�1h1(h02)�1h2 · · · (h0n�1)
�1hn�1.
Then
h0nh�1n 2 H1H2 · · ·Hn�1 \Hn = {e} =) h0nh
�1n = e =) h0n = hn.
We can thus cancel hn and h0n from opposite sides of (?) and repeat the precedingto get h0n�1 = hn�1. continuing, we eventually get hi = h0i for i = 1, 2, . . . , n.
9. NORMAL SUBGROUPS AND FACTOR GROUPS 131
Now define � : G ! H1 �H2 � · · ·�Hn by
�(h1h2 · · ·hn) = (h1, h2, · · · , hn).
From the above, � is well-=defined. If �(h1h2 · · ·hn) = �(h01h02 · · ·h0n),
(h1, h2, · · · , hn) = (h01, h02, · · · , h0n) =)
hi = h0i for i = 1, . . . , n =) h1h2 · · ·hn = h01h02 · · ·h0n,
so � is 1–1. That � is onto is clear.
[To show operation preservation.]
Now let h1h2 · · ·hn, h01h02 · · ·h0n 2 G. Again, using the commutativity shown
above,
�⇥�(h1h2 · · ·hn)(h
01h02 · · ·h0n)
⇤= �
⇥(h1h
01)(h2h
02) · · · (hnh
0n)
⇤=
(h1h01, h2h
02, · · · , hnh
0n) = (h1h2 · · ·hn) · (h01h
02 · · ·h0n) =
�(h1h2 · · ·hn)�(h01h02 · · ·h0n),
so operations are preserved and � is an isomorphism. ⇤
Note. If G = H1 �H2 � · · ·�Hn, then for
Hi = {e}�Hi � {e}, i = 1, ..n,
G = H1 ⇥H2 ⇥ · · ·Hn.
Clearly, each Hi ⇡ Hi.
132 9. NORMAL SUBGROUPS AND FACTOR GROUPS
Theorem (9.7 – Classification of Groups of Order p2). Every group oforder p2, where p is a prime, is isomorphic to Zp2 or Zp � Zp.
Proof.
Let G be a group of order p2, p a prime. If G has an element of order p2, thenG ⇡ Zp2. Otherwise, by Corollary 2 of Lagrange’s Theorem, we may assumeevery non-identity element of G has order p.
[To show that 8 a 2 G, hai is normal in G.] Suppose this is not the case. Then9 b 2 G 3�� bab�1 62 hai. Then hai and hbab�1i are distinct subgroups of orderp. Since hai \ hbab�1i is a subgroup of both hai and hbab�1i,hai \ hbab�1i = {e}. Thus the distinct left cosets of hbab�1i are
hbab�1i, ahbab�1i, a2hbab�1i, . . . , ap�1hbab�1i.Since b�1 must lie in one of these cosets,
b�1 = ai(bab�1)j = aibajb�1
for some i and j. Cancelling the b�1 terms, we get
e = aibaj =) b = a�i�j 2 hai,a contradiction. Thus, 8 a 2 G, hai is normal in G.
Now let x be any non-identity element in G and y any element of G not in hxi.Then, by comparing orders and from Theorem 9.6,
G = hxi ⇥ hyi ⇡ Zp � Zp.
⇤
Corollary. If G is a group of order p2, where p is a prime, then G isAbelian.
9. NORMAL SUBGROUPS AND FACTOR GROUPS 133
Example. We use Theorem 8.3, its corollary, and Theorem 9.6 for thefollowing.
If m = n1n2 · · ·nk where gcd(ni, nj) = 1 for i 6= j, then