FIRST-YEAR GROUP THEORY 1 DEFINITIONS AND EXAMPLES It is highly desirable that you buy, or in some other way have access to a copy of, the following book which will be referred to in these notes as Jordan and Jordan": Groups" by C. R. Jordan and D. A. Jordan, published by Edward Arnold in 1994, ISBN 0-340- 61045-x. CONVENTIONS: Unless stated to the contrary, the following symbols will be used to denote the given sets: C for the set of complex numbers; R for the set of real numbers; Q for the set of rational numbers; Z for the set of integers; Z n or Z=nZ for the set of integers mod(n); M n (A) for the set of n n matrices with entries in the set A. A binary operation on a non-empty set A is a way of starting from any two elements a and b (in that order) of A and producing from them a unique corresponding element a b of A (Jordan and Jordan, Section 4.2). In practice in this course such an operation will usually be either addition or multiplication on a suitable set. Informally, a binary operation is a way of combining any two things of a certain type (i.e. elements of A) to produce an answer of the same type (i.e. an element ofa). DEFINITION 1.1: (Group; Jordan and Jordan, Section 4.3) A non-empty set G is said to be a group under or with respect to the operation, or(g; ) is said to be a group, if all of the four following axioms are satisfied: G1 (Closure) is a binary operation on G, i.e. to each ordered pair (a; b) of elements of G there corresponds a unique element a b of G (and we say that G is closed under or with respect to ); G2 (Associativity) For all a, b, c 2 G we have (a b) c = a (b c); G3 (Existence of identity element) There is an element e of G such that e g = g e = g for all g 2 G, and this element e is called the identity element (or sometimes the neutral element) of G; G4 (Existence of inverses) Corresponding to each element g of G there is an element h of G such that g h = h g = e; and this element h is called the inverse of g and is often denoted by g 1. DEFINITION 1.2: (Abelian) A group G as in 1.1 is said to be Abelian if a b = b a for all a, b 2 G. DEFINITION 1.3: (Order of a group) We shall use ord(g) to denote the number of elements in a group G and shall call ord(g) the order of G. Thus if G has a finite number n of elements then we say that G has order n and write ord(g) =n; if G has an infinite number of elements then we say that G has infinite order or is infinite.
REMARKS 1.4: (a) In general discussion the operation in a group G is usually written as multiplication and a b is written as ab. (b) If the operation in a group G is multiplication then we often write 1 rather than e for the identity element. (c) If the operation in a group G is addition then we usually write 0 rather than e for the identity element, and a rather than a 1 for the inverse of an element a of G. Thus in additive notation and using 0 for the identity element we have 0+g = g +0= g for all g 2 G; and given g 2 G there exists h 2 G such that g + h = h + g = 0, and we denote this element h by g. Also we write a b as a shorthand for a +( b). Additive notation is very rarely used for groups which are not Abelian. (d) The notation ord(g) is not standard. The notation jaj is often used to denote the number of elements in a set A whether or not A is a group. (e) The identity element e of a group (G; ) is unique in the sense that if f is also an element of G such that f g = g f = g for all g 2 G then e = f; to justify this note that because e is an identity element then e f = f, and because f is an identity element wehave e f = e. (f) If g is an element of a group G then the inverse of g is unique (Jordan and Jordan, Section 4.5). EXAMPLE 1.5: Let G consist of the four complex numbers 1, i, 1, i where i 2 = 1. Then G is a group with respect to multiplication of complex numbers. It is routine to check that the product of any two elements of G is again an element of G (Closure). In examples like this we shall take the associative law for granted because it is well-known that multiplication of complex numbers is always associative. The element 1 of G is the identity element of G. We can check directly that each elementofg has an inverse in G: we have 1:1 = 1 so that 1 is its own inverse; i: i = i:i = 1 so that i is the inverse of i and equally well i is the inverse of i; and 1: 1=1so that 1 isitsown inverse. We have ord(g) =4and G is Abelian. EXAMPLE 1.6: Let G be the set of non-zero rational numbers. Then G is a group under the usual multiplication of rational numbers. Recall that a rational number is a number of the form a=b where a and b are integers with b 6= 0. So a non-zero rational number is of the form a=b where both a and b are non-zero integers. To check that G is closed under multiplication, let g, h 2 G. Then g = a=b and h = c=d where all of a, b, c, d are non-zero integers. We have gh = ac=bd where ac and bd are non-zero integers, so that gh 2 G. Thus G is closed under multiplication. We shall take it for granted that the multiplication in G is associative. The identity element of G is the rational number 1=1 = 1. If g = a=b as above then the inverse of g is b=a which is again an element ofg. Therefore G is a group under multiplication, and clearly G is infinite and Abelian. EXAMPLE 1.7: We can proceed as in 1.6 to show that the set of positive rational numbers is an infinite Abelian group under multiplication. EXAMPLE 1.8: The set G of negative rational numbers is not a group under multiplication because closure fails. For instance 1, 2 2 G but 1: 2=262 G. EXAMPLE 1.9: The set Q of all rational numbers is not a group under multiplication because there is no inverse for the element 0;clearly the identity element is 1, and there is no element h of Q such that 0:h =1.
Similarly we get groups by taking all positive real numbers, all non-zero real numbers, or all non-zero complex numbers under multiplication. These groups are all infinite Abelian. EXAMPLE 1.10: The non-zero integers do not form a group under multiplication (Exercise). EXAMPLE 1.11: The positive integers do not form a group under multiplication (Exercise). REMARKS 1.12: Let n be a positive integer. The elements of the set Z n or Z=nZ of integers mod(n) are conveniently represented by the symbols 0, 1, 2,..., n 1, and these symbols are added and multiplied in the usual way except that multiples of n can be replaced by 0. Similarly any two integers which differ by amultiple of n, i.e. whose difference is divisible by n, are considered to be the same as elements of Z=nZ. For instance, working in Z=7Z the elements can be denoted 0, 1, 2, 3, 4, 5, 6; we have 3+4=7=0,4+6=10=3,4:6 =24=21+3=3,3:4 =12=5, and so on. Note that Z=nZ has n distinct elements. We shall always take it for granted that addition and multiplication in Z=nZ are associative. Note that we doallow n = 1 in which case Z=1Z consists of the single element 0. EXAMPLE 1.13: Let G be the set of non-zero elements of Z=5Z. Then G is a group under multiplication. The elements of G will be denoted by the sumbols 1, 2, 3, 4. To check closure is routine but tedious: for instance 3, 3 2 G and 3:3 = 9 = 4 2 G; also 2, 4 2 G and 2:4 = 8 = 3 2 G. We can state without proof in examples like this that the multiplication is associative. Clearly 1 is the identity element of G. To check for inverses note that in G we have 1 = 1:1 = 2:3 = 4:4 so that 1 and 4 are their own inverses, and 2 and 3 are inverses for each other. EXAMPLE 1.14: Let G consist of the non-zero elements of Z=6Z. Then G is not a group under multiplication. For instance 2, 3 2 G but 2:3 = 6 = 0 62 G, so that closure fails. REMARKS 1.15: (a) The key difference between Examples 1.13 and 1.14 is that 5 is a prime number but 6 is not. (b) When working in Z=nZ we can often shorten calculations by using negative symbols rather than positive ones. For instance in Z=5Z we have 4= 1, so that 4:4 = 1: 1=1. This trick" becomes more useful when n is large. For instance, working in Z=103Z it would be rather tedious without a calculator to work out 96:97 directly; but 96 = 7 and 97 = 6 in Z=103Z, sothat 96:97 = 7: 6 = 42. NOTATION 1.16: (Jordan and Jordan, Section 4.5) Let G be a group in which the operation is multiplication, so that we write xy rather than x y. It follows from the associative law that brackets can be inserted or deleted in products of three or more elements of G (provided that we do not change the order of the factors). For instance, if x, y, z, t 2 G then we have (x(yz))t = (xy)(zt) and we would usually just write this as xyzt. There is no logical problem about using the usual notation for positive integer powers, e.g. using x 2 for xx, x 3 for xxx, and so on. For x 2 G we define x 0 to be the identity element ofg, and for any positive integer n we define x n to be (x 1 ) n. For instance we define x 2 to be the square of x 1 ; it is an exercise to show that x 2 is then also the inverse of x 2. For all x 2 G and all i, j 2 Z we have x i x j = x i+j and (x i ) j = x ij.
WARNING 1.17: Let x and y be elements of a multiplicatively-written group G. Then (xy) 2 is not always the same as x 2 y 2. All that we can say in general about (xy) 2 is that it is xyxy, unless for some special reason we have xy = yx in which case (xy) 2 = x 2 y 2. Similarly for (xy) 3, etc. EXAMPLE 1.18: The set G of all non-zero elements of Z=7Z is a group under multiplication. We could check closure directly, but it will follow from something else which we are about to do. We take associativity for granted. The identity element is 1. Set x = 3. Recall that in G we have for instance 9 = 2 becuase 7 divides 9 2. Thus in G we have x = 3, x 2 = 9 = 2, x 3 = x 2 :x = 2:3 = 6 = 1, x 4 = x 3 :x = 1:3 = 3, x 5 = x 4 :x = 3:3 = 9 = 2, x 6 = x 5 :x = 2:3 = 6 =1,x 7 = x 6 :x =1:x = x, x 7 = x 6 :x = x 2, and so on. Hence G consists of the powers of x with x 6 =1. It follows that G is closed under multiplication. We can write the elements of G as 3, 2, 1, 1, 2, 3 rather than as 1, 2,..., 6 where for instance 5 = 2. The inverse of 1 is 1; the inverse of 1 is 1; 2 and 3 are inverses for each other because 2: 3 = 3:2 = 6 = 1; and similarly 3 and 2 are inverses for each other. Therefore G is a group under multiplication. NOTE: In the next example, and in a few other places in these notes, we shall mention matrices. You may ignore this material if you are not familiar with matrix theory, and you will not be asked about matrices, but this material is of great mathematical significance and is included for the benefit of those who know about matrices (see Jordan and Jordan, Section 3.1). EXAMPLE 1.19: The set M 2 (R) of all real 2 2 matrices is not a group under matrix multiplication because for instance there is no inverse for the zero matrix. Let G be the set of elements of M 2 (R) which have non-zero determinant. Then G is a group under matrix multiplication, and G is non-abelian (i.e. not Abelian); this group is usually denoted by either GL 2 (R) or GL(2;R) where GL" stands for general linear". Similarly, given a fixed positive integer n, the group GL(n; R) consists of all n n real matrices with non-zero determinant under matrix multiplication. EXAMPLE 1.20: (Symmetries of a circle; Jordan and Jordan, Section 1.2) Think of a circle, for instance the so-called unit circle which is the circle of radius 1 with centre at the origin in the usual (x; y)-coordinate system for the plane. It may help to think of the circle as being a rigid circular wire without any marks on it. Let G be the set of all the ways of moving the circle, without bending, stretching, tearing, etc., which put the circle back into its original position. There are two types of thing you can do: you can rotate the circle through some angle about its centre; or you can turn the circle over through 180 degrees about one of its diameters (which is equivalent to reflecting the circle in that diameter). We make the important convention that when we say Reflect the circle in the diameter D" we think of D as being a fixed line through the origin in the (x; y)-plane (for instance D might be the x-axis) and D does not move when we move the circle. Thus G consists of rotations and reflections, and if a and b are elements of G we define ab to bethe effect of doing b first and then doing a. It can be checked that G is a group under this operation. The identity element 1 or e of G is the transformation of the circle which leaves all its points fixed. Let a be a rotation through t degrees. Then the inverse of a is rotation through t degrees (i.e. rotation through t degrees in the opposite direction). By convention, angles of rotation are usually measured anti-clockwise. If b is a reflection in a diameter (equivalently, rotation through 180 degrees about that diameter) then b 2 = 1 (i.e. doing b twice takes every point onthecircle back to where it started from). Thus every reflection is its own inverse. It is crucial to note that if a is a rotation about the centre and b is a reflection as above, then bab = a 1. For instance if a is rotation through t degrees about the centre and b is reflection in the x-axis, then the equation bab = a 1 says that the combined effect of reflecting in the x-axis
and then rotating through t degrees and then again reflecting in the x-axis is the same as simply rotating through t degrees in the opposite direction. Because b 1 = b, the equation bab = a 1 can be rewritten as b 1 ab = a 1 or ab = ba 1. Because we can have a 6= a 1 (for instance if a is rotation through 30 degrees) it follows that we can have ab 6= ba, so that G is not Abelian. This group G is known as the group of symmetries of the circle, or as the orthogonal group O 2 (R) or just O 2. EXAMPLE 1.21: (Symmetries of an equilateral triangle; the dihedral group D 3 ) Using the same ideas as in 1.20 we take G to be the set of symmetries of an equilateral triangle T, and for two elements x and y of G we use xy to mean do y first and then do x". Then G is a group with respect to this multiplication, and we shall denote the six elements of G as follows: e or1isthe identity element (i.e. the transformation which leaves every point oft fixed); a is rotation about the centre through 120 degrees anti-clockwise; a 2 is a done twice, so that 2 is rotation through 240 degrees; b, c, d are the reflections in (i.e. turning T over through 180 degrees around) a line joining one vertex of T to the mid-point of the opposite side. We have b 2 = c 2 = d 2 = e, so that each of b, c, d is its own inverse; also a 3 is rotation through 360 degrees so that a 3 = e and a 2 is the inverse of a, i.e. a 2 = a 1 ; and as in 1.20 we have bab = a 1, i.e. ab = ba 1. In fact G is a non-abelian group of order 6, and a convenient way of writing the elements of G is as follows: G consists of the six distinct elements e, a, a 2, b, ab, a 2 b with a 3 = e = b 2 and ab = ba 1. Thus the reflections can be written as b, ab, a 2 b rather than just as b, c, d. Some typical calculations with this algebraic notation are: (ab) 2 = abab = (ab)ab = (ba 1 )ab = b(a 1 a)b = beb = b 2 = e, or with greater experience we would do this more quickly as (ab) 2 = abab = ba 1 ab = b 2 = e; also ab:a 2 b = ba 1 a 2 b = bab = a 1 = a 2. This group G is known as the group of symmetries of the equilateral triangle or as the dihedral group of order 6andis denoted by D 3. EXAMPLE 1.22: (The symmetries of a square; the dihedral group D 4 ) Proceeding as in 1.21 let G be the group of symmetries of a square S. Then G has eight elements which can be denoted as follows: e is the identity element; a is rotation through 90 degrees anti-clockwise; a 2 is rotation trhough 180 degrees; a 3 is rotation through 270 degrees, so that a 3 = a 1 and a 4 = e; x is the reflection in the diagonal joining the top left corner of S to the bottom right corner if you have S oriented with sides horizontal and vertical (and this diagonal must be thought of as fixed in space and not moving with the square); y is reflection in the other diagonal; z is reflection in the line joining the midpoints of the top and bottom edges; t is reflection in the line joining the midpoints of the left and right edges. In fact if we take b to be any one of the reflections x, y, z, or t then it is convenient to list the elements of G as e, a, a 2, a 3, b, ab, a 2 b, a 3 b with a 4 = e = b 2 and ab = ba 1. Then G is a non-abelian group of order 8 and the usual notation for G is D 4. EXAMPLE 1.23: (The dihedral group D n ) Let n be a positive integer. The dihedral group D n is a group of order 2n whose elements can be denoted by e, a, a 2,..., a n 1, b, ab, a 2 b,..., a n 1 b where e is the identity element and a n = e = b 2 and ab = ba 1. When n = 1 we have a = e, and d 1 consists of just e and b with b 2 = e. When n = 2 we have a 2 = e so that a 1 = a and ab = ba 1 = ba. In fact D 2 is Abelian of order 4 with elements e, a, b, ab where a 2 = b 2 = (ab) 2 = e. Now suppose that n 3. Then D n is not Abelian and it can be thought of as the group of symmetries of a regular n-sided polygon. Geometrically, ifn is odd there are n reflection-lines each joining a vertex to the midpoint of the opposite side; whereas if n is even there are n=2 reflection-lines joining midpoints of opposite sides, and n=2 reflection-lines joining pairs of opposite vertices. For more information about groups of symmetries see Jordan and Jordan, Section 5.3.
EXAMPLE 1.24: Let G consist of the four following ordered pairs of integers: e =(1; 1); a = (1; 1); b = ( 1; 1); c = ( 1; 1). We define multiplication in G componentwise, by which we mean that the product of pairs (x; y) and (z; t) is defined to be the pair (xz; yt) formed by multiplying corresponding components. Then G is an Abelian group of order 4 under this multiplication. We have a 2 = b 2 = c 2 = e where e is the identity element, and ab = c. EXAMPLE 1.25: Working in Z=8Z let G consist of the four elements 1, 3, 5, 7. Then G is an Abelian group of order 4 under multiplication. We have 3 2 = 9 = 1 recalling that we are working in Z=8Z, 5 2 = 25 = 1, and 7 2 = 49 = 1 (alternatively, 7 2 =( 1) 2 = 1). If we put e =1, a =3,b =5,c =7then we have a 2 = b 2 = c 2 = e and ab =3:5 =15=7=c. Thus in abstract terms this group seems to be just like the one in 1.24, even though they are not literally the same group. DEFINITION 1.26: (Klein 4-group V ) The Klein 4-group V is an Abelian group of order 4 in which the square of every element isthe identity. If (G; ) isany group of order 4 with identity element e and x x = e for all x 2 G, then we say that G is the Klein 4-group. So far we have met the Klein 4-group in three different settings, namely as D 2 in 1.23, and as the groups in 1.24 and 1.25. ADDITIVE NOTATION: Let G be a group in which the operation is written as addition. Then we usually denote the identity element of G by 0, so that 0+x = x +0 = x for all x 2 G. The inverse of an element x of G is denoted by x rather than x 1, so that x +( x) = ( x) +x = 0. We write x y rather than x +( y), and x + y rather than ( x) +y. For an integer n and an element x of G we write nx rather than x n ; thus 3x = x + x + x, and 3x = (3x) = (x + x + x) = x x x =3( x). Additive notation is almost never used for groups which are not Abelian. EXAMPLE 1.27: The sets Z, Q, R, C, Z=nZ are all groups under addition. EXAMPLE 1.28: The positive integers do not form a group under addition because there is no identity element (note that 0 is not positive). EXAMPLE 1.29: The non-negative integers 0, 1, 2,... do not form a group under addition. There is an identity element, namely 0. But, for instance, there is no additive inverse for 2 (note that 2 is not allowed because it is not non-negative). EXAMPLE 1.30: Let n be a fixed integer. Then the set of all integers which are divisible by n is a group under addition. PROPOSITION 1.31: (Cancellation laws) Let (G; ) be a group, and let x, y, z 2 G such that either x y = x z or y x = z x. Then y = z. PROOF: Suppose that x y = x z. Let e be the identity element ofg and let a be the inverse of x. Thus a x = x a = e. Multiplying each side of the equation x y = x z on the left by a gives a (x y) =a (x z), i.e. (a x) y =(ax) z, i.e. e y = e z, i.e. y = z. The case in which y x = z x is done similarly by multiplying on the right by a. COROLLARY 1.32: Let (G; ) be a group, and let g 2 G such that g g = g. Then g = e, where e is the identity element of G. PROOF: Because e is the identity element we have g e = g. Therefore g g = g e, so that g = e by 1.31. CAYLEY TABLES: If a group (G; ) has a small number of elements, it is possible to display the values of g h for g, h 2 G as a table. The rows and columns of the table are labelled by the symbols which denote the elements of the group, and the value of g h is put in the table where the rog labelled g meets the column labelled h. Such tables are known as Cayley tables.
For more information see Jordan and Jordan, Sections 1.1 and 4.5; it will be asssumed that you have read these. EXAMPLE 1.33: (how to form the group U n ) Let n be a positive integer. We shall find the largest multiplicative group U n of non-zero elements of Z=nZ. You should learn the following procedure for doing this. One way of denoting the elements of Z=nZ is to use the symbols 0, 1, 2,..., n 1; and another is to use the symbols for all the integers but to define two integers a and b to be equal as elements of Z=nZ if and only if n divides a b. Then U n consists of all the elements of Z=nZ which, as integers, are relatively prime to n (the justification for this statement is given in 1.34). For instance the elements of Z=5Z are 0, 1, 2, 3, 4; we can for present purposes always ignore 0; the remaining elements 1, 2, 3, 4 are all relatively prime to 5, so that U 5 consists of all the non-zero elements of Z=5Z as a group under multiplication (see 1.13). More generally, if p is any prime number then U p consists of all the non-zero elements of Z=pZ (note that U p has order p 1). On the other hand, if n is not prime then U n does not consist of all the non-zero elements of Z=nZ. For instance the non-zero elements of Z=12Z are 1, 2, 3, 4,..., 10, 11; of these only 1, 5, 7, 11 are relatively prime to 12; note that in Z=12Z we have 7= 5 and 11 = 1; hence U 12 consists of the four elements ±1 and ±5 under multiplication mod(12). In fact U 12 is another instance of the Klein 4-group. See Jordan and Jordan, Section 8.5. PROPOSITION 1.34: Let a and n be integers with n positive. Then a has a multiplicative inverse in Z=nZ if and only if a and n are relatively prime. PROOF: Firstly suppose that a and n are relatively prime, i.e. that hcf(a; n) = 1. Then sa + tn = 1 for some s, t 2 Z. Thus sa 1 is divisible by n, so that in Z=nZ we have sa = 1 and s is a multiplicative inverse for a. Conversely suppose that a has a multiplicative inverse in Z=nZ. Then there is an integer x such that xa =1inZ=nZ, i.e. such that the integer xa 1 is divisible by n. Hence xa 1=ny for some y 2 Z. Therefore xa ny = 1, so that any integer which divides both a and n must also divide 1. But the only integers which divide 1 are ±1. Therefore a and n are relatively prime. EXAMPLE 1.35: (The quaternion group) Let G consist of the eight 2 2 complex matrices which are the four following elements of M 2 (C) TOGETHER WITH THEIR NEGATIVES: E = 1 0 0 1 i 0 K = 0 i 0 1 L = 1 0 0 i M = i 0 Thus G consists of the eight elements E, K, L, m, E, K, L, and M. It can be shown that G is a group under matrix multiplication, known as the quaternion group. To show that G is closed under multiplication involves elementary but slightly tedious checking; for instance direct calculation gives that K 2 = E and KL = M. We can take associativity for granted because matrix multiplication is always associative. Clearly E is the identity element of G. It can be checked that E and E are their own inverses; K and K are inverses for each other; L and L are inverses for each other; M and M are inverses for each other. This group G and the dihedral group D 4 are essentially the only two non-abelian groups of order 8. ; ; ; :