GROUPS AND APPLICATIONS

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1 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA GROUPS AND APPLICATIONS Tadao ODA Tohoku Uiversity, Japa Keywords: group, homomorphism, quotiet group, group actio, trasformatio, symmetry, represetatio Cotets. Groups 2. Commutative Groups 3. Examples 4. Subgroups 5. Homomorphisms 6. Quotiet Groups 7. Homomorphism ad Isomorphism Theorems 8. Cyclic Groups 9. Direct Products 0. Fiitely Geerated Abelia Groups. Group Actios ad Symmetry 2. Solvable Groups 3. Represetatios of Fiite Groups Glossary Bibliography Biographical Sketch Summary Groups are sets with oe algebraic operatio. Their basic properties are explaied with may typical examples. The otio of groups with simple but very abstract form historically grew out of more cocrete trasformatios ad turs out to be extremely powerful thaks to the very abstractess. Results o matrices ad liear algebra i Matrices, Vectors, Determiats ad Liear Algebra will be freely used.. Groups A group is a set G edowed with a map G G ( x, y) xy G called the group operatio (alteratively, group law, or multiplicatio) satisfyig the followig coditios: (Associativity) ( xy) z= xyz ( ) holds for all x, yz, G. (Existece of the uity) There exists e G, called the uity, such that xe= ex= x holds Ecyclopedia of Life Support Systems (EOLSS) 52

2 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA for ay x G. (Existece of the iverse) For ay x G there exists such that xx = x x =e. x G, called the iverse of x, Whe G is a fiite set with g elemets, G is called a fiite group of order g. The otatio g =: G is ofte used. Otherwise, G is called a ifiite group. 2. Commutative Groups A group G is said to be commutative (alteratively, Abelia) if the followig coditio is satisfied: (Commutativity) xy = yx holds for all x, y G. The group operatio for a commutative group G is ofte deoted as a additio G G ( x, y) x+ y G so that the coditios are of the followig form: (Associativity) ( x + y) + z = x+ ( y+ z) holds for all x, yz, G. (Existece of the zero) There exists 0 G, called the zero, such that x + 0= 0+ x= x holds for ay x G. (Existece of the mius) For ay x G there exists x G, called the mius of x, such that x+ ( x) = ( x) + x = 0. (Commutativity) x + y= y+ x holds for all x, y G. 3. Examples The set = {, 2,, 0,, 2, } of itegers uder the usual additio is a ifiite commutative group. The subset {, } of uder the usual multiplicatio is a commutative group of order 2. The subset {,, i, i} of complex umbers uder the usual multiplicatio is a commutative group of order 4. The subset {,, i, i, j, j, k, k} H of Hamilto s quaterios uder the usual multiplicatio is a o-commutative group of order 8, ad is called the quaterio group. The sets,, ad H of ratioal umbers, real umbers, complex umbers ad Hamilto s quaterios, respectively, are commutative groups uder the usual additio. The set := {0} of ozero ratioal umbers is a group uder the usual multiplicatio. Similarly, := {0}, := {0}, H := H {0} are groups uder the usual multiplicatio. Ecyclopedia of Life Support Systems (EOLSS) 53

3 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA Let X be a set. The the set Aut( X ) of oe-to-oe ad oto maps f : X X of X to itself is a group uder the compositio f f of f, f Aut( X ). The idetity map id : X X is the uity, while the iverse map f is the iverse of f Aut( X ). I the previous example, let X := {, 2, 3,, }. The S := Aut({, 2,,}) is usually called the symmetric group of degree. It has order S =!, ad cosists of the permutatios of {, 2,, }. The set GL ( ) of real square ivertible matrices of size is a group uder the matrix multiplicatio, ad is called the geeral liear group. The idetity matrix is the uity, while the iverse matrix A is the iverse of A GL. I Similarly, GL ( ) (resp. GL ( C) ) is the group of ratioal (resp. complex) square ivertible matrices of size. Note that o quaterioic aalog of these groups exists. Note further that if X :=, for istace, i the example above, the GL is a subset of 4. Subgroups Aut( ) cosistig of those oe-to-oe ad oto maps which are liear, that is, which preserve the vector space structure of cosistig of the real colum vectors of size. A oempty subset H G of a group G is called a subgroup of G if xy H holds for all x, y H. It is easily see that H itself is a group uder the operatio iduced by that of G, sice it cotais the uity e G, is closed uder the group operatio of G ad y H holds for all y H. (Here is how to show these facts: There exists at least oe x0 H sice H is assumed to be oempty. Hece y = ey H for ay y H by defiitio. Cosequetly, all x, y H.) Sometimes, the otatio H e= x x is i H. Moreover, 0 0 xy x( y ) H = holds for < G is used to mea that H is a subgroup of G. The itersectio H H of subgroups H < G ad H < G is a subgroup of G. Oe obviously has H H < H as well as H H < H. A subgroup N of G is said to be ormal if xyx x G. It is coveiet to describe this fact as xnx := { xyx y N}. N holds for all ad all xnx y N = for ay x G It is coveiet to deote N G to say that N is a ormal subgroup of G. N, where Oe obviously has N N G for N G ad N G. Obviously, N N N ad N N N hold. Moreover, N H H for ay subgroup H < G. However, a Ecyclopedia of Life Support Systems (EOLSS) 54

4 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA ormal subgroup M of a ormal subgroup N G G. Namely, M N G does ot imply M G. eed ot be a ormal subgroup of Note that subgroups of a commutative group are automatically ormal. Note also that {} e ad G itself are ormal subgroups of ay group G. Here are examples of subgroups ad ormal subgroups: Ay of the additive groups i the sequece H is a subgroup of those to the right. Ay of the multiplicative groups i the sequece H is a subgroup of those to the right. The multiplicative group {, } is a subgroup of the multiplicative group. The multiplicative group {,,, i i} is a subgroup of the multiplicative group, ad cotais the multiplicative group {, } above as a subgroup. The quaterio group {,, i, i, j, j, k, k} is a subgroup of the multiplicative group H, ad cotais the multiplicative group {,,, i i} above as a subgroup. Subgroups of S s are called permutatio groups. The set SL ( ) of real square matrices A of size with det( A) = is a ormal subgroup of GL ( ), called the real special liear group. The ormality is a easy cosequece of det( PAP ) = det( P)det( A)det( P) = det( A). The ormal subgroups SL GL ( ) ad SL GL are defied similarly. The set of positive real umbers is a subgroup of the multiplicative group >0 of ozero real umbers. The orthogoal group O ( ) is the subgroup of GL cosistig of orthogoal matrices. The uitary group U( ) is the subgroup of GL cosistig of uitary matrices. Oe has O ( ) = U ( ) GL. A subgroup of the additive group ca be show to be either {0} or, for each positive iteger, the subset := { a a } = {, 2,, 0,, 2, } of the multiples of. 5. Homomorphisms A homomorphism from a group G to aother group G is a map f : G G such that f ( xy) = f ( x) f ( y) for all x, y G. It is easily see that f () e coicides with the uity e G, ad f ( x ) = f( x) for ay x G. A oe-to-oe homomorphism is also called a ijective homomorphism or a ijectio, while a oto homomorphism is called a surjective homomorphism or a surjectio. A isomorphism is a homomorphism that is both oe-to-oe ad oto. Ecyclopedia of Life Support Systems (EOLSS) 55

5 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA A homomorphism of a group G to itself is called a edomorphism of G. A isomorphism of a group G to itself is called a automorphism of G. The image f ( G) := { f( x) x G} of a homomorphism f : G G is easily see to be a subgroup of G. More importatly, the kerel ker( f ) := { x G f( x) = e } is a ormal subgroup of G, sice f ( yxy ) = f( y) f( x) f( y) = f( y) ef( y) = e holds for ay x ker( f ) ad ay y G. Here are examples: det : GL sedig A to its determiat det( A) is a (surjective) homomorphism. By defiitio, ker(det) = SL ( ). Likewise, surjective homomorphisms det : GL ad det : GL have respective kerel SL ( ) ad SL. The sigature homomorphism sg : S {, } from the symmetric group S to the multiplicative group {, } is defied as follows: Cosider the polyomial called the differece product i variables x, x,, x defied by 2 Δ (,,, ):= ( ). x x 2 x xi x j < i j For each permutatio σ S, the variables are permuted by σ ad Δ ( x, x,, x,, x ) = sg( ) Δ ( x, x,, x,, x ) σ () σ(2) σ( i) σ ( ) σ 2 i with sg( σ ) =±. The kerel A := ker(sg) is called the alteratig group of degree. The map : > 0 sedig z to its absolute value z is a homomorphism. Its kerel is the set U() := {exp( iθ) = cosθ + isi θ θ } of complex umbers with absolute value. The kerel SO( ) of det : O ( ) { ± } is called the group of rotatios. Obviously, SO( ) = O( ) SL ( ). The kerel SU ( ) of de t : U( ) U() is called the special uitary group. Obviously, SU ( ) = U ( ) SL ( ) as well as SO( ) = SU ( ) GL ( ). For ay elemet a of a group G, a map Ia : G G is defied to be the oe sedig x G to I ( ) a x := axa. It is a automorphism of G sice I ( ) ( ) a xy = a xy a = ( axa )( aya ) = Ia( xi ) a( y) with I as the iverse map. I a is called a ier a Ecyclopedia of Life Support Systems (EOLSS) 56

6 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA G Iab Ia Ib automorphism of. Note that ab( ) ( ) ( ) ( ) a ( b ( )) ( a b)( ) I x = ab x ab = a bxb a = I I x = I I x. 6. Quotiet Groups For a subgroup H of a group =, sice G ad x G, the subset xh := { xy y H} of G called the left coset of x G with respect to H. Oe calls x a represetative of the coset xh. Note that elemets of the form xy 0 for y 0 H are represetatives of the coset xh as well. Likewise, Hx := { yx y H} is called the right coset of x G with respect to H, while x is called a represetative of the coset. Elemets of the form y0x are represetatives of Hx as well. (I the case of additive groups, the otatio x + H :={ x+ y y H} is used.) The set G/ H of all the left cosets xh i G with respect to H is called the left coset space of G with respect to a subgroup H. Likewise, the set H \G is the set of all the right cosets Hx i G with respect to H ad is called the right coset space of G with respect to a subgroup H. The left coset space ad the right coset space are also called homogeeous spaces of G with respect to a subgroup H. [G: H] is defied to be the umber of cosets i G with respect to H, ad is called the idex of H i G. It does ot matter whether the left cosets or the right cosets are couted. Ideed, the oe-to-oe ad oto set map G G sedig x G to x G maps each left coset xh to the right coset Hx iducig a oe-to-oe ad oto set map GH / H\G. is Whe G is a fiite group, the formula G = [ G: H] H due to Lagrage turs out to be very useful, where G ad H are the orders of the groups G ad H, respectively. Of special iterest are the coset spaces with respect to ormal subgroups. For a ormal subgroup N G of a group G, the left coset ad the right coset of ay x G coicide so that xn = Nx by the defiitio of ormality. Moreover, the coset space G/ N turs out to be a group, called the quotiet group with respect to N uder the followig group law: The product of cosets xn ad x N is defied to be ( xn)( xn ):= xxn, which is well-defied idepedetly of the choice of represetatives. Ideed, let xy ad x y with yy, N be other represetatives. The Ecyclopedia of Life Support Systems (EOLSS) 57

7 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA xy xy N = xx x yx yn = xxn, sice ( )( ) (( ) ) en = N is the uity of GN /, while ( x ) yx N by ormality. The coset x N is easily see to be the iverse of xn. The map π : G G/ N sedig x G to π ( x) := xn is called the projectio, ad is easily see to be a oto homomorphism with kerel ker( π ) = N. The quotiet groups produce iterestig ew groups. 7. Homomorphism ad Isomorphism Theorems A homomorphism of groups f G G : iduces a isomorphism f : G/ ker( f) f( G), which is kow as the homomorphism theorem. For ormal subgroups N ad N of a group G with N N, oe has a atural surjective homomorphism f : GN / GN / sedig xn to xn. Its kerel is ker( f ) = N / N, so that by the above theorem the so-called first isomorphism theorem ( GN / )/( N / N) GN / holds. Ay subgroup of G/ N is of the form H/ N for a subgroup H of G cotaiig N. Moreover, H/ N is ormal i G/ N if ad oly if H is ormal i G. I geeral, let H be a subgroup of a group G ad N a ormal subgroup of G. The the set HN := { h h H, N} is easily see to be a subgroup of G cotaiig N as a ormal subgroup. (I the case of additive groups, the otatio H + N :={ h+ h H, N} is used.) The projectio π : G G/ N iduces a homomorphism π H : H GN / by restrictio. Its kerel is ker( π H ) = H N, while its image is the quotiet group HN/ N. Thus the so-called secod isomorphism theorem H/ ( H N) HN/ N holds. These simple iocuous lookig theorems tur out to be extremely useful as show later. The map f : U() sedig θ to f( θ ) := exp(2 πθ i ) U() is a surjective homomorphism with kerel ker( f ) =. Thus a isomorphism Ecyclopedia of Life Support Systems (EOLSS) 58

8 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA / U (). It was see earlier that the subgroups of the additive group are {0} ad for positive itegers. For positive itegers ad m, oe has m if ad oly if m divides. Deote by gcd( m, ) ad lcm( m, ) the greatest commo divisor ad the least commo multiple of m ad, respectively. The m = lcm( m, ) ad m + = gcd( m, ). The well-kow formula lcm( m, ) gcd( m, ) = m is the secod isomorphism theorem i disguise. The subgroups of / for a positive iteger are m / for the divisors of. m 8. Cyclic Groups Let x be a elemet of a group G. The a homomorphism fx : G is defied to be the map sedig m to xx x ( m times) if m > 0 m fx ( m) := x := e if m= 0. x x x ( mtimes) if m < 0 Its kerel could be either {0} or for a positive iteger as was see before. Its image is deoted by f ( ) x =: x ad is called the subgroup of G geerated by x. x G is said to be of ifiite order if ker( f x) = {0}. I this case, the image x of f x cosists of mutually distict m x s for all m ad oe has a isomorphism 2 2 x = { x,, x, exx,,, }. Ay of these isomorphic groups is called a ifiite cyclic group. x G is said to be of order if ker( fx) = for a positive iteger. I this case, the image cosists of distict elemets ad oe has a isomorphism / x = e, x, x,, x. 2 { } Ecyclopedia of Life Support Systems (EOLSS) 59

9 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA Bibliography Arti, M. (99). Algebra, xviii+68 pp. Pretice Hall, Ic., Eglewood Cliffs, NJ, ISBN [This is oe of the advaced comprehesive textbooks o algebra suitable for further study.] Birkhoff, G. ad Mac Lae, S. (965). A Survey of Moder Algebra, Third Editio, x+437 pp. The Macmilla Co., New York; Collier-Macmilla Ltd., Lodo [A stadard textbook o algebra.] Birkhoff, G. ad Mac Lae, S. (988). Algebra, Third Editio, xx+626 pp., Chelsea Publishig Compay, New York, ISBN [A classic comprehesive textbook o algebra. Origially published from Macmilla, Ic.] Lag, S. (2002). Algebra, Revised Third Editio, xvi+94 pp.,graduate Texts i Mathematics 2, Spriger-Verlag, New York, ISBN X[A advaced comprehesive textbook o algebra Ecyclopedia of Life Support Systems (EOLSS) 77

10 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA suitable for graduate studets. Origially published from Addiso-Wesley Publishig Compay.] Roa, M. (2006). Symmetry ad the Moster. Oe of the Greatest Quests of Mathematics, vi+25 pp. Oxford Uiversity Press, Oxford, ISBN ; [A book iteded primarily for a o-mathematical audiece, explais the history of groups sice the aciet Greek period till the recet classificatio of fiite simple groups.] Biographical Sketch Tadao ODA, bor 940 i Kyoto, Japa Educatio: BS i Mathematics, Kyoto Uiversity, Japa (March, 962). MS i Mathematics, Kyoto Uiversity, Japa (March, 964). Ph.D. i Mathematics, Harvard Uiversity, U.S.A. (Jue, 967). Positios held: Assistat, Departmet of Mathematics, Nagoya Uiversity, Japa (April, 964-July, 968) Istructor, Departmet of Mathematics, Priceto Uiversity, U.S.A. (September, 967-Jue, 968) Assistat Professor, Departmet of Mathematics, Nagoya Uiversity, Japa (July, 968-September, 975) Professor, Mathematical Istitute, Tohoku Uiversity, Japa (October, 975-March, 2003) Professor Emeritus, Tohoku Uiversity (April, 2003 to date) Ecyclopedia of Life Support Systems (EOLSS) 78