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(c) Define an action of G on S by left multiplication, aT = {at:t T} for a∈ G and

T ∈ S. Prove that this is a group action. (d) Provep|OT|for someT ∈ S.

(e) Let{T1, . . . , Tu} be an orbit such thatpu andH ={g∈G:gT1 =T1}. Prove that

H is a subgroup of Gand show that |G|=u|H|. (f) Show thatpk divides|H|and pk≤ |H|.

(g) Show that |H|=|OT| ≤pk; conclude that therefore pk=|H|.

26. Let Gbe a group. Prove thatG′ =⟨aba−1b−1 :a, b∈G⟩ is a normal subgroup of G andG/G′ is abelian. Find an example to show that{aba−1b−1:a, b∈G}is not necessarily a group.

15.4 A Project

The main objective of finite group theory is to classify all possible finite groups up to isomorphism. This problem is very difficult even if we try to classify the groups of order less than or equal to60. However, we can break the problem down into several intermediate problems. This is a challenging project that requires a working knowledge of the group theory you have learned up to this point. Even if you do not complete it, it will teach you a great deal about finite groups. You can use Table 15.21 as a guide.

Order Number Order Number Order Number Order Number

1 ? 16 14 31 1 46 2 2 ? 17 1 32 51 47 1 3 ? 18 ? 33 1 48 52 4 ? 19 ? 34 ? 49 ? 5 ? 20 5 35 1 50 5 6 ? 21 ? 36 14 51 ? 7 ? 22 2 37 1 52 ? 8 ? 23 1 38 ? 53 ? 9 ? 24 ? 39 2 54 15 10 ? 25 2 40 14 55 2 11 ? 26 2 41 1 56 ? 12 5 27 5 42 ? 57 2 13 ? 28 ? 43 1 58 ? 14 ? 29 1 44 4 59 1 15 1 30 4 45 ? 60 13

Table 15.21: Numbers of distinct groups G,|G| ≤60

1. Find all simple groupsG ( |G| ≤ 60). Do not use the Odd Order Theorem unless you are prepared to prove it.

2. Find the number of distinct groupsG, where the order of Gis nforn= 1, . . . ,60. 3. Find the actual groups (up to isomorphism) for each n.

References and Suggested Readings

[1] Edwards, H. “A Short History of the Fields Medal,”Mathematical Intelligencer1(1978), 127–29.

[2] Feit, W. and Thompson, J. G. “Solvability of Groups of Odd Order,”Pacific Journal of Mathematics 13(1963), 775–1029.

[3] Gallian, J. A. “The Search for Finite Simple Groups,”Mathematics Magazine49(1976), 163–79.

[4] Gorenstein, D. “Classifying the Finite Simple Groups,” Bulletin of the American Mathematical Society 14(1986), 1–98.

[5] Gorenstein, D.Finite Groups. AMS Chelsea Publishing, Providence RI, 1968. [6] Gorenstein, D., Lyons, R., and Solomon, R. The Classification of Finite Simple

16

Rings

Up to this point we have studied sets with a single binary operation satisfying certain ax- ioms, but we are often more interested in working with sets that have two binary operations. For example, one of the most natural algebraic structures to study is the integers with the operations of addition and multiplication. These operations are related to one another by the distributive property. If we consider a set with two such related binary operations sat- isfying certain axioms, we have an algebraic structure called a ring. In a ring we add and multiply elements such as real numbers, complex numbers, matrices, and functions.

16.1

Rings

A nonempty set R is a ring if it has two closed binary operations, addition and multipli- cation, satisfying the following conditions.

1. a+b=b+a fora, b∈R.

2. (a+b) +c=a+ (b+c) fora, b, c∈R.

3. There is an element 0in R such thata+ 0 =afor alla∈R.

4. For every element a∈R, there exists an elementainR such that a+ (−a) = 0. 5. (ab)c=a(bc) fora, b, c∈R.

6. Fora, b, c∈R,

a(b+c) =ab+ac

(a+b)c=ac+bc.

This last condition, the distributive axiom, relates the binary operations of addition and multiplication. Notice that the first four axioms simply require that a ring be an abelian group under addition, so we could also have defined a ring to be an abelian group (R,+)

together with a second binary operation satisfying the fifth and sixth conditions given above. If there is an element 1∈R such that 1̸= 0 and 1a=a1 =a for each elementa∈R, we say thatR is a ring withunity oridentity. A ringR for whichab=bafor alla, binR is called a commutative ring. A commutative ringR with identity is called anintegral domain if, for every a, b ∈R such that ab= 0, either a = 0or b = 0. A division ring is a ring R, with an identity, in which every nonzero element in R is a unit; that is, for each a ∈R witha ̸= 0, there exists a unique element a−1 such that a−1a=aa−1 = 1. A commutative division ring is called afield. The relationship among rings, integral domains, division rings, and fields is shown in Figure 16.1.

Rings with Identity Division Rings Commutative Rings Integral Domains Rings Fields

Figure 16.1: Types of rings

Example 16.2. As we have mentioned previously, the integers form a ring. In fact, Z is an integral domain. Certainly if ab = 0 for two integers a and b, either a = 0 or b = 0. However, Z is not a field. There is no integer that is the multiplicative inverse of 2, since

1/2is not an integer. The only integers with multiplicative inverses are1 and 1.

Example 16.3. Under the ordinary operations of addition and multiplication, all of the familiar number systems are rings: the rationals,Q; the real numbers, R; and the complex numbers, C. Each of these rings is a field.

Example 16.4. We can define the product of two elementsaand b inZn by ab (modn).

For instance, in Z12, 5·7 11 (mod12). This product makes the abelian group Zn into a ring. Certainly Zn is a commutative ring; however, it may fail to be an integral domain.

If we consider 3·4 0 (mod12) in Z12, it is easy to see that a product of two nonzero

elements in the ring can be equal to zero.

A nonzero element ain a ring R is called a zero divisor if there is a nonzero element bin R such thatab= 0. In the previous example,3and 4 are zero divisors in Z12.

Example 16.5. In calculus the continuous real-valued functions on an interval[a, b]form a commutative ring. We add or multiply two functions by adding or multiplying the values of the functions. If f(x) =x2 and g(x) =cosx, then(f+g)(x) =f(x) +g(x) =x2+cosx and (f g)(x) =f(x)g(x) =x2cosx.

Example 16.6. The2×2matrices with entries inRform a ring under the usual operations of matrix addition and multiplication. This ring is noncommutative, since it is usually the case thatAB̸=BA. Also, notice that we can have AB= 0 when neitherAnor B is zero. Example 16.7. For an example of a noncommutative division ring, let

1 = ( 1 0 0 1 ) , i= ( 0 1 1 0 ) , j= ( 0 i i 0 ) , k= ( i 0 0 −i ) , wherei2 =1. These elements satisfy the following relations:

i2 =j2=k2 =1

ij=k jk=i ki=j

16.1. RINGS 183 kj=i

ik=j.

Let H consist of elements of the form a+bi+cj+dk, where a, b, c, d are real numbers. Equivalently,Hcan be considered to be the set of all 2×2 matrices of the form

(

α β

−β α )

,

where α = a+di and β = b+ci are complex numbers. We can define addition and multiplication onH either by the usual matrix operations or in terms of the generators 1, i,j, and k: (a1+b1i+c1j+d1k) + (a2+b2i+c2j+d2k) = (a1+a2) + (b1+b2)i+ (c1+c2)j+ (d1+d2)k and (a1+b1i+c1j+d1k)(a2+b2i+c2j+d2k) =α+βi+γj+δk, where α=a1a2−b1b2−c1c2−d1d2 β =a1b2+a2b1+c1d2−d1c2 γ =a1c2−b1d2+c1a2+d1b2 δ=a1d2+b1c2−c1b2+d1a2.

Though multiplication looks complicated, it is actually a straightforward computation if we remember that we just add and multiply elements in H like polynomials and keep in mind the relationships between the generatorsi,j, andk. The ring His called the ring of quaternions.

To show that the quaternions are a division ring, we must be able to find an inverse for each nonzero element. Notice that

(a+bi+cj+dk)(a−bicjdk) =a2+b2+c2+d2.

This element can be zero only if a,b,c, and dare all zero. So if a+bi+cj+dk̸= 0,

(a+bi+cj+dk) ( a−bi−cj−dk a2+b2+c2+d2 ) = 1. Proposition 16.8. Let R be a ring with a, b∈R. Then

1. a0 = 0a= 0;

2. a(−b) = (−a)b=−ab; 3. (−a)(−b) =ab.

Proof. To prove (1), observe that

a0 =a(0 + 0) =a0 +a0;

hence, a0 = 0. Similarly, 0a = 0. For (2), we have ab+a(−b) = a(b−b) = a0 = 0; consequently, ab = a(−b). Similarly, ab = (−a)b. Part (3) follows directly from (2) since(−a)(−b) =(a(−b)) =(−ab) =ab.

Just as we have subgroups of groups, we have an analogous class of substructures for rings. A subring S of a ring R is a subset S of R such that S is also a ring under the inherited operations from R.

Example 16.9. The ring nZis a subring of Z. Notice that even though the original ring may have an identity, we do not require that its subring have an identity. We have the following chain of subrings:

ZQRC.

The following proposition gives us some easy criteria for determining whether or not a subset of a ring is indeed a subring. (We will leave the proof of this proposition as an exercise.)

Proposition 16.10. Let R be a ring and S a subset of R. Then S is a subring of R if and only if the following conditions are satisfied.

1. =∅.

2. rs∈S for all r, s∈S. 3. r−s∈S for all r, s∈S.

Example 16.11. LetR =M2(R) be the ring of 2×2 matrices with entries in R. If T is

the set of upper triangular matrices inR; i.e., T = {( a b 0 c ) :a, b, c∈R } , thenT is a subring ofR. If A= ( a b 0 c ) and B = ( a′ b′ 0 c′ )

are inT, then clearly A−B is also inT. Also, AB= ( aa′ ab′+bc′ 0 cc′ ) is in T.