O Ifiite Groups that are Isomorphic to its Proper Ifiite Subgroup Jaymar Talledo Baliho Abstract Two groups are isomorphic if there exists a isomorphism betwee them Lagrage Theorem states that the order of a subgroup divides the order of the group This implies that a fiite group is ot isomorphic to its proper subgroup Some ifiite groups which are isomorphic to some of its proper subgroup are illustrated i this paper Moreover, it is show through cocepts i liear algebra that these groups really exist 1 Itroductio This chapter presets the backgroud of the study, objectives of the study, sigificace of the study, methodology ad basic cocepts i group theory Backgroud of the Study May of the studets takig itroductory abstract aim to kow whether a certai group is isomorphic to a proper subgroup of itself Fiite groups are ot isomorphic to a proper subgroup of themselves sice two groups havig differet cardialities caot be isomorphic The additive group Z of all itegers is isomorphic to the subgroup 2 Z of eve itegers via the map f : Z 2Z give by f ( x) 2x, for all Z Also, the multiplicative group Q of all positive ratioal is isomorphic to some of its proper subgroup usig the fuctio f : Q Q defied 3 by f ( x) x, for all x Q These groups show that ideed, there are ifiite groups which are isomorphic to some of their proper subgroups But beig ifiite is ot 23
eough because some ifiite groups are ot isomorphic to their proper subgroups Hece, this paper characterizes some ifiite groups which are isomorphic to its proper subgroup Objectives of the Study This paper will ivestigate those ifiite groups which are isomorphic to its proper subgroup Specifically, this paper aims to show the followig: 1 The additive group is isomorphic to its proper subgroup Z, where Z {0,1, 1} 2 The additive group of all ratioal umbers is ot isomorphic to ay of its proper subgroup 3 The additive group Q Z is isomorphic to its proper subgroup Q Z, where Z {0,1, 1} 4 Suppose that ad are groups, oe of which is isomorphic to some of its proper subgroup The G His also isomorphic to some of its proper subgroup 5 Let G1, G2,, G be groups If for some i{1,2,3,, }, is isomorphic to its proper subgroup the G1 G2 G is also isomorphic to its proper subgroup Sigificace of the Study G Z Q If a group is isomorphic to its proper subgroup ad this subgroup has a kow structure, the the structure of the group ca also be determied via this isomorphism Also, if oe group is kow to be isomorphic to its proper subgroup, the ay group isomorphic to this group will also be isomorphic to its proper subgroup Thus, this paper which ivolves basic cocepts of group theory is of great importace to mathematics ethusiast H G i 24
2 Methodology This paper exposes the work of Shau Fallat, Chi Kwo Li, David Lutzer, ad David Staford etitled O Groups That Are Isomorphic to a Proper Subgroup Defiitios, examples ad prelimiary cocepts are beig preseted i order to provide detailed proof of the mai results Prelimiaries This sectio cotais defiitios ad prelimiary cocepts that are eeded for further uderstadig of the study 11 Basic Defiitios ad Kow Results Defiitio 151 (Hugerford, 1980): A fuctio f : X Y is a relatio betwee X ad Y with the property that each x X appears as the first member of exactly oe ordered pair ( x, y) f Such a fuctio is called a map or mappig of X itoy Defiitio 152 (Hugerford, 1980): Let f : X Y be a fuctio, ad let be a subset of ad B be a subset ofy The image of i Y uder f is the set f [ A] { f ( a) : a A} The set f[x] is called the rage of A f The iverse image of B X is the set X A f 1 [ B] { x X : f ( x) B} Defiitio 153 (Maclae, Birkhoff, 1967): A fuctio f : X Y is i well defied if a bimplies f(a) = f(b) ii ijective if a, a ' A, f(a) = f(b) implies a = b iii surjective if for each b B, b = f(a) for some a A iv bijective if it is both ijective ad bijective Defiitio 154 (Fraleigh, 1994): (Priciple of Mathematical Iductio): Let P () be a statemet cocerig the positive iteger Suppose that 1 P(1) is true, ad 25
2 If P (k) is true, the P (k+1) is true The P () is true Z Defiitio 155 (Fraleigh, 1994): Let G be oempty set with a biary operatio The G is called a group if the followig axioms hold: [G1]: a bg, a, b G (closure property) [G2]: (a b) c = a (b c), a, b, c G (associative property) [G3]: There is a elemet e i G such thata G,ea a a e(existece of a idetity elemet i G) 1 [G4]: Correspodig to each a G, there is a elemet a G such 1 1 that a a a a e (existece of iverses i G) Defiitio 156 (Fraleigh, 1994): The order of a group G, deoted, is the umber of elemets i G A group with fiite order is called a fiite group Otherwise, it is a fiite group by G Defiitio 157 (Fraleigh, 1994): Let G be a group If H G ad is a group uder the biary operatio, the H is a subgroup of G writte H G H ig Defiitio 158 (Fraleigh, 1994): If G is a group, the the subgroup cosistig of G itself is called a improper subgroup ofg All the other subgroups of G are called proper subgroups Defiitio 159 (Fraleigh, 1994): If G is a group, a is i G ad N, the a is a product of factors each equal to a; that is, a aaa a( - factors) Defiitio 1510(Fraleigh, 1994): Let G be group If a G, the { a : } is called the cyclic subgroup of G geerated by a ad will be deoted by a If G b for someb G, the G is said to be cyclic 26
Defiitio 1511(Fraleigh, 1994): Let G be e group ad a G The order of a, deoted a, is the smallest positive iteger such that The order of the cyclic subgroup is equal to the order of a, that is, <a> = a a a e Theorem 1512 (Hugerford, 1980): Every ifiite cyclic group is isomorphic to the additive group Z ad every fiite cyclic group of order m is isomorphic to the additive group Z Theorem 1513 (Fraleigh, 1994): Lagrage s Theorem: Let H be subgroup of a fiite group G The the order of H divides the order of G I particular, if a G, the a divides G Defiitio 1514 (Fraleigh, 1994): Let G1, G2,, G be groups Cosider 1 Gi G1 G2 G {( g1, g2,, g) : gi G} Defie a biary operatio o by ( g, g,, g ) ( h, h,, h ) ( g h, g h,, g h ), 1 2 1 2 1 1 2 2 G i gi, hi G The G1 G2 G is group called the direct sum of the groups uder the biary operatio + G i Defiitio 1515 (Fraleigh, 1994): Let G ad H be groups A fuctio f : G H is a homomorphism provided that f (a*b) = f (a)*f (b) a, b G If f is ijective, the f is called a isomorphism I this case, G ad H are said to be isomorphic writteg H The kerel of f, deoted by ker f, is the subgroup f 1 ({ e'}) where e is the idetity elemet of G Defiitio 1516 (Fraleigh, 1994): Let ad d be o-zero itegers The d divides, deoted by d/, if there exist a iteger q such that = dq 27
3 Results ad Discussios This chapter presets the isomorphism of group oto its proper subgroup for the case where the group is a direct sum 31 Direct Sum of Groups This sectio tells us whe the direct sum of groups is isomorphic to some of its proper subgroup Theorem 311 The additive group of itegers is isomorphic to some of its proper subgroup, where \{0,1, 1} Proof: Let f : be defied by f ( a) a, a Let a, b If a b, the f ( a) a b f ( b) Thus, f is well-defied Suppose f ( a) f ( b) The, a b which implies that a b Hece, f is oe-to-oe Also f ( a b) ( a b) a b f ( a) f ( b) Hece, f is a homomorphism Let b The b m for some m Take a m The f ( a) a m b So, f is oto Therefore, the additive group is isomorphoc to the proper subgroup The followig corollary follows from Theorem 311 ad Theorem 1512 Corollary 312 Every ifiite cyclic group is isomorphic to some of its proper subgroup Theorem 313 The additive group of all ratioal umbers is ot isomorphic to ay of its proper subgroup 28
Proof: Suppose that f : is a ozero homomorphism Note that ad Claim 1: f ( x) f (1) x, x, where f (1) 0 Let x The x a for some a, b, b 0 Observe that b a a a bf ( x) bf f f b b b a a f b b a f b b f( a) 1 f (1) f( a 1) f (1 1) f (1) f (1) f (1) af (1) f(1) a a Thus, f ( x) f (1) f (1) x b Moreover, f(1) 0 sice f is a ozero homomorphism Claim 2: f is oe-to-oe Let a, b with f ( a) f ( b) The f (1) a f (1) b by Claim 1 Sice f(1) 0 ad f( ) is a subgroup of, cacellatio holds, that is, a b Hece, f is oe-to-oe Claim 3: f is oto a Let x The x with a, b, b 0 Sice 0 f (1), there exist b 29
ozero itegers c ad b such that bc 0sice b 0 ad 0 c c f (1) d This implies that ad c ad a Let t The t ad f ( t) f (1) t x Thus, bc d bc b f is oto Therefore, is a isomorphism This meas that every homomorphism f : is a isomorphism Thus, is ot isomorphic to ay of its proper subgroup f Observe that the additive group of itegers is isomorphic to some of its proper subgroup but the additive group of ratioal umbers is ot What ca be said about? Theorem 314 The additive group is isomorphic to some if its proper subgroup, where \{0,1, 1} Proof: Note that is a proper subgroup of sice (1,1) but (1,1) Let f : be defied by f (( s, a)) ( s, a) Let ( s, a) ad ( t, b) be i with ( s, a) ( t, b) The s t ad a b Now f (( s, a)) ( s, a) ( t, b) f (( t, b)) Thus f is well-defied Suppose that f (( s, a)) f (( t, b)) The, ( s, a) ( t, b) implies that s t ad a b this shows that ( s, a) ( t, b) ad is oe-tooe Let x The, x ( s, a) for some s, a, Take ( sa, ) The, f (( s, a)) ( s, a) implyig that is oto Therefore, isomorphic to its proper subgroup Theorem 314 is geeralized i the followig theorem Theorem 315 Suppose that G ad H are groups, oe of which is isomorphic to some of its proper subgroup The G His also isomorphic to its proper subgroup f f 30
Proof: Sice e G G 1 1 Let G ad H 1 G1 G such that a isomorphism :G G 1 are groups, it follows that the idetity elemets say ad eh H exist Thus, ( eg, eh) G1 H ad sog 1 H ( g, h ),( g, h ) G H Now, Let 1 1 2 2 1 ( g, h ) ( g, h ) ( g g, h h ) G H sice G ad H are closed 1 1 2 2 1 2 1 2 1 uder additio Thus, G1 H is also closed uder + Note that ( g, h ) ( e, e ) ( g e, h e ) ( g, h ) 1 1 G1 H 1 G1 1 H 1 1 Thus ( eg, e ) 1 H is the idetity elemet ig 1 H Also, ( e, e ) ( g, h ) ( g, h ) ( g g, h h ) implies G1 H 1 1 2 2 1 2 1 2 that g g e ad h h e 1 2 G1 1 2 H Thus g e g g ad 2 G1 1 1 h2 eh h1 h1 Hece (g 2h2 ) ( g1, h1 ) G1 H is the iverse (g 1, h1 ) G1 H Thus, G1 H is a subgroup ofg H Cosider the fuctio :G H G1 H defied by (( g, h)) ( ( g), h), ( g, h) G H Let ( g, h),( g ', h') G H If ( g, h) ( g ', h'), the g g ' ad h h' So, (( g, h)) ( ( g), h) ( ( g '), h') ( g ', h') well-defied Observe that, (( g, h) ( g ', h')) ( g g ', h h') Thus ( ( g g '), h h ') ( ( g) ( g '), h h ') ( ( g), h) ( ( g '), h ') (( g, h) (( g ', h')) Thus, is a homomorphism Now, if (( g, h)) (( g ', h')), the ( ( g), h) ( ( g '), h') so that ( g) ( g ') ad h h' Sice is oe-tooe, g g' Thus, ( g, h) ( g ', h') ad is oe-to-oe Let ( g1, h) G1 H Sice g 1 G 1 ad is oto, g G such that ( g) g1 Cosider ( g, h) G H The, (( g, h)) ( ( g), h) ( g1, h) 31
Thus, is a isomorphism ad subgroup G 1 H G H is isomorphic to its proper Corollary 316 Let G1, G2,, G be groups If for some i {1,2,, }, Gi is isomorphic to some of its proper subgroup, the G1 G2 G is also isomorphic to its proper subgroup Proof: If oe of G1 or G2is isomorphic to its proper subgroup, the by Theorem 315, G1 G2is isomorphic to its proper subgroup Assume that G1 G2 Gk is isomorphic to its proper subgroup for all k 1 The by Theorem 315, G1 G2 Gk 1 ( G1 G2 Gk ) Gk 1 is isomorphic to some of its proper subgroup Therefore, by PMI, G G G is also isomorphic to some of its proper subgroup 1 2 4 Summary, Coclusio ad Recommedatios This chapter summarizes the results beig studied i this paper ad presets some recommedatios for further iquiries Summary ad Coclusio This paper obtaied the followig results o groups which are isomorphic to some of its proper subgroup: 1 The additive group of itegers is isomorphic to some of its proper subgroup, where \{0,1, 1} (Theorem 311) 32
2 Every ifiite cyclic group is isomorphic to some of its proper subgroup (Corollary 312) 3 The additive group of all ratioal umbers is ot isomorphic to ay of its proper subgroup (Theorem 313) 4 The additive group is isomorphic to some if it s proper subgroup, where \{0,1, 1} (Theorem 314) H 5 Suppose that G ad are groups, oe of which is isomorphic to some of its proper subgroup The G His also isomorphic to its proper subgroup (Theorem 315) 6 Let G1, G2,, G be groups If for some i {1,2,, }, Gi is isomorphic to some of its proper subgroup, the G1 G2 G is also isomorphic to its proper subgroup (Theorem 316) Recommedatios The author recommeds the followig questios for further ivestigatio 1 If G His isomorphic to some of its proper subgroup, will it follow that G or H is isomorphic to some of its proper subgroup? 2 How may homomorphism ad isomorphism exist from the additive group ad ito itself? 33
3 Oe ca show that the usual fields are ot isomorphic to proper subfields of themselves but there are fields lyig betwee them that are isomorphic to proper subfields of themselves Which fields are field isomorphic to proper subfields of themselves? ad ad List of Refereces Fraleigh, JB A First Course i Abstract Algebra 1994 Addiso Wesley Publishig Compay Ic Hugerford, T Algebra 1980 New York: Spriger-Verlag Maclae, S ad Birkhoff, G Algebra 1967 New York: Collier- Macmilla LIST OF NOTATIONS H G H is a subgroup of G H G H is a proper subgroup of G for all there exists empty set A B Direct sum of sets A ad B set of atural umber group of itegers 2 group of eve itegers group of ratioal umbers group of real umbers group of itegers modulo isomorphic to 34