SOME APPLICATIONS FROM STUDENT S CONTESTS ABOUT THE GROUP OF AUTOMORPHISMS OF Z

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1 Joural o Sciece ad Arts Year 5 No () ORIGINAL PAPER SOME APPLICATIONS FROM STUDENT S CONTESTS ABOUT THE GROUP OF AUTOMORPHISMS OF GEORGIANA VELICU ELENA CATALINA GRINDEI Mauscrit received: 6005; Acceted aer: 80505; Published olie: Abstract This article is a collectio o a ew iterestig alicatios o iite grous ad morhisms o grous alicatios which already have bee give to udergraduate s iteratioal ad atioal cotests some o them usig oly basic ideas o moder ad abstract algebra Studets studyig algebra i school will obviously go urther as will those who are iterested i oe or more o the alicatios The aim o this aer is to sustai studets studyig algebra ad imrove their ability to hadle abstract ideas Keywords: grou geerator automorhism INTRODUCTION I mathematics a grou is a algebraic structure more seciically a oemty set edowed with a low o comositio which combie betwee them two elemets obtaiig i this way a third elemet o the cosidered set But to become a grou the set with its low o comositio must satisy some coditios amed the axioms o the grou: associability the existece o the eutral elemet ad the existece o the symmetric o ay elemet rom the cosidered set Although these roerties called axioms are commo or may algebraic structures like the sets o umbers articularly the set o ositive itegers the ormulatio o these axioms is well detached by the ature o the grou ad rom its law The existece o the grous i may domais o mathematics but ot oly makes them a ricile o orgaizatio i cotemorary mathematics The cocet o grou aeared i coectio with the study o the olyomial equatios study made by the Frech mathematicia Évariste Galois i the years 80 Ater the cotributios obtaied rom other domais like umber theory ad geometry the otio o grou was geeralized i the 70 Studyig grous mathematicias develoed several otatios to slit them i smaller arts easier to uderstad ad study like subgrous ad simle grous A theory o grous was develoed or the iite grous which culmiated with the classiicatio o simle ad iite grous i 98 The classiicatio o all iite grous is a very diicult work By Lagrage s theorem all iite grous o order where is a rime ositive iteger are cyclic grous ad obviously commutative grous which are deoted (u to a isomorhism) by It is easy to rove that every grou o order is also a commutative oe but this assertio ca t be geeralized to grous o order ; we have i this sese the examle o the dihedral grou D Uiversity Valahia o Targoviste Faculty o Scieces ad Arts 00 Targoviste Romaia georgiaavelicu@yahoocom Uiversity Valahia o Targoviste Faculty o Scieces ad Arts 00 Targoviste Romaia cataelly@yahoocom ISSN: Mathematics Sectio

2 58 Some alicatios rom Georgiaa Velicu Elea Catalia Gridei o order 8 = which is ot a commutative oe Some algorithms ca be used to geerate lists o small grous but we do t have yet a classiicatio o all iite grous A itermediate ste was the classiicatio o the iite ad simle grous We remid that a otrivial grou is a simle grou i it s oly ormal subgrous is itsel ad the trivial subgrou meas the subgrou geerated by the eutral elemet Jorda-Hölder s theorem resets the simle grous like the costituets o all iite grous Makig a list o all iite ad simle grous was a great achievemet i the grou s theory ad it is due to Richard Borcherds laureate o Fields Medal i 998 or the mostrous mooshie cojecture a amazig coectio betwee the largest soradic grou iite ad simle ad the moster grou o order M = = We also have to seciy that the morhisms ad isomorhisms are very imortat i the study o iite grous Remid that i algebra a isomorhism (i Greek: isos = "equal" ad morhe = "orm") is a alicatio betwee two sets equied with a algebraic structure alicatio which satisies two coditios: is a morhism (meas that the algebraic structure is reserved) ay isomorhism have a iverse which is also a morhism (i :A B is a isomorhism the does exist g:b A also a morhism so that g = A ad g = B ) I this way ay two algebraic structures with the roerty that does exist betwee them a isomorhism are called isomorhe As a cosequece ay two algebraic structure beig isomorhe have the same roerties A secial class o isomorhisms is the class o the automorhisms o a grou A automorhism o a grou G is a isomorhism rom G to G The set o all automorhism o G is deoted by Aut(G) Next we reset some imortat alicatios related to the automorhisms o a imortat grou: the additive grou o classes modulo- AUTOMORPHISMS FOR First we remid some roerties o the automorhisms o ( + ) ad ext we reset a way to determie the umber o the automorhisms o the additive grou where is a rime umber Remak A class aˆ = / is a geerator or the additive grou i ad oly i the great commo divisor o a ad is ( a ) = I cosequece the umber o the geerators or is give by the Euler s uctio ϕ () Proo First we have: i aˆ is a geerator or = < aˆ > the there exist a elemet u so that ˆ = u aˆ ua (mod ) ua v ad we obtai ua v = ( a ) = Iverse we have: Also or ay class ( a ) = u v the au + v = so au (mod ) au = ˆ b ˆ we have > b ˆ = ˆ bˆ = au bˆ = bu aˆ < aˆ the we have < a ˆ >= wwwjosaro Mathematics Sectio

3 Some alicatios rom Georgiaa Velicu Elea Catalia Gridei 59 Examle The additive grou = have ( ) ϕ = = 8 geerators ad the set o geerators is Ge ) = { kˆ / ( k) = } = {ˆ5ˆ7ˆ79 } ( Remak I aˆ = / is a geerator or the ϕ : ( ˆ) = ˆ ϕ Aut( ) Proo It is obvious that ϕ is a morhism o grous: ϕ k ka k is a automorhism o ϕ ( k+ k) = ϕ( k + k) = ( k + k) a = ka+ ka = ϕ( k) + ϕ( k) k k The ijectivity o ϕ : ϕ( k) = ϕ( k) k k k a = ka ka ka (mod ) ( k k ) a 0 (mod ) but â is a geerator or so ( a ) = ad so we obtai = k (mod k ) k k Because ϕ is deied ro to obvious that ϕ is a bijectio Fially we have that ϕ is a automorhism o ad it is ijective it is Examles By the revious remark we have: Aut( ) = ; Aut( ) ; Aut( ) ; Aut( ) is a rime umber Proo For we obtai oly oe automorhism: id : id ( kˆ) kˆ = so Aut ( ) = Also the additive grou have two geerators: ˆ ad ˆ ad or ˆ we have the idetity automorhism id ad or ˆ we have ϕ : ( kˆ) = k ϕ with ϕ( 0ˆ) = 0ˆ ϕ(ˆ) = ˆ ad ϕ(ˆ) = ˆ I the same way the grou have two geerators: ˆ ad ˆ ad so two automorhisms: the idetity ma id ad ϕ : ( kˆ) = k ϕ with ϕ ( 0ˆ) = 0ˆ ϕ(ˆ) = ˆ ϕ(ˆ) = ˆ adϕ(ˆ) = ˆ I geeral or a rime umber the coditio ( a ) = imlies a { } so have geerators ad or every such a geerator we obtai a automorhism o ϕ : ϕ( kˆ) = ka a { } totally automorhisms ad Aut ( ) Below we reset the tables or the ollowig grous: ( Aut ( ) ) ad ( + ) rom where we observe that they are isomorhe: ( Aut ( ) ) id ϕ id ϕ id ϕ ϕ id ( ) 0ˆ ˆ 0ˆ 0ˆ ˆ ˆ ˆ 0ˆ A iterestig questio about the automorhisms o a grou is: which o the automorhisms admits ixed oits? For that we have the ollowig airmatio: Remark The grou where is a rime umber it s a grou i which ay automorhism other tha the idetity is a automorhism who admit a ixed oit ISSN: Mathematics Sectio

4 60 Some alicatios rom Georgiaa Velicu Elea Catalia Gridei Proo Let cosider Aut ) We have: ( kˆ) = akˆ a { } I have at least two ixed oits the ( k ˆ 0ˆ so that ( kˆ) = kˆ the a kˆ = kˆ ( a ) k ˆ = 0ˆ a but a { } the a = Fially we have: ( kˆ) = k ˆ = Cosequece Aut ) U ( ) ( Proo Aut ( = = ˆ ) { : / ( xˆ) x (ˆ) x ad < (ˆ) >= } Aut( ) = { : / ( xˆ) = x (ˆ) xˆ ad (ˆ) = kˆ ( k ) = } the Aut ) U ( ) much more we have Aut( ) = ϕ( ) where ϕ () is Euler s uctio ( We saw above that the set o all automorhisms o the additive grou is well deied We are iterested ow to determie the grou o automorhisms o the Cartesia roduct m N m which is also a additive grou We kow that there exist m a isomorhism m m whe m N m are two ositive itegers with ( m ) = Aother imortat result is: Aut( ) S A simle roo o the existece o this isomorhism is: i we cosider the set aˆ bˆ A = / aˆ bˆ cˆ dˆ ( ˆ 0ˆ ˆ 0ˆ) ( ˆ 0ˆ ˆ 0ˆ) ( ˆ ˆ ˆ a or b ad c or d ad a c or b dˆ) ˆ cˆ d ˆ 0ˆ ˆ 0ˆ 0ˆ ˆ 0ˆ ˆ ˆ ˆ ˆ 0ˆ ˆ ˆ A = 0ˆ ˆ ˆ ˆ ˆ 0ˆ ˆ ˆ ˆ 0ˆ ˆ ˆ 0ˆ ˆ the it is obvious that A S ad ϕ : Aut( ) S deied by ϕ( ) = aˆ bˆ where cˆ dˆ ( 0ˆˆ) = ( aˆ bˆ) ad ( ˆ0ˆ) = ( cˆ dˆ ) is a isomorhism o grous the we have Aut( S ) Next we will seciy the umber o automorhisms o where is a rime umber ad how they are deied o the geerators It is obvious that the idetity ma is a automorhism ad also because is a commutative grou we have that its iterior automorhisms coicide with the idetity ma Remember that: i ( G ) is a grou ad a G is a elemet rom G the alicatio ϕ a : G G a g = a g a ϕ ( ) g G is a automorhism o G called the iterior automorhism o G Alicatio I is a rime umber the Aut( ) = ( ) (Berkeley 00) wwwjosaro Mathematics Sectio

5 Some alicatios rom Georgiaa Velicu Elea Catalia Gridei 6 Solutio It is obvious that a automorhism o images o the geerators ( 0) ad 0 ) ( is well determiated by the So the ollowig ma: : (0) = ( a b ) where ( a b ) ( c d ) (0 ) = ( c d ) is a automorhism i ad oly i a ad a / b c ad c / ad also d ( ) * The direct imlicatio is obvious because i is a automorhism o with ( 0) = ( a b ) ad ( 0 ) = ( c d ) the ( a b ) is ot caceled by but ( c d ) must be caceled by ad so we obtai that multile o ( a b ) = ( a0) the d 0 a / ad c / More ( c d ) ca t be a The reverse imlicatio: cosiderig two elemets ( a b ) ad ( c d ) havig the above roerties we have a morhism g ( 0) = ( a b ) ad g ( 0 ) = ( c d ) because ( a b ) is caceled by coditio a / imlies that ord ( a b ) = The air ( d ) g : so that ad ( d ) c by The c ca t be a multile o ( a b ) = ( a0) because d ( ) * The we have by the Lagrage s theorem ) > But g( the act that ) = the g is a surjective morhism ad cosiderig g( is a iite grou we obtai g also ijective the g is a automorhism Now let s cout the airs ( a b ) ad ( c d ) with the above roerties: or a we have ossibilities or b we have ossibilities or c we have ossibilities or d we have ossibilities totally there are ( ) ( ) = ( ) automorhisms o Examles: For = the grou have 8 automorhisms I the case o = the grou 9 have 08 automorhisms ad or = have 000 automorhisms Next we reset the table or = ad we seciy all it s automorhisms ad subgrous Let s coside = {( x y) / x y } ISSN: Mathematics Sectio

6 6 Some alicatios rom Georgiaa Velicu Elea Catalia Gridei + ( 00) ( 0 ) ( 0) ( ) ( 0) ( ) ( 0) ( ) ( 00) ( 00) ( 0 ) ( 0) ( ) ( 0) ( ) ( 0) ( ) ( 0 ) ( 0 ) ( 00) ( ) ( 0) ( ) ( 0) ( ) ( 0) ( 0) ( 0) ( ) ( 0) ( ) ( 0) ( ) ( 00) ( 0 ) ( ) ( ) ( 0) ( ) ( 0) ( ) ( 0) ( 0 ) ( 00) ( 0) ( 0) ( ) ( 0) ( ) ( 00) ( 0 ) ( 0) ( ) ( ) ( ) ( 0) ( ) ( 0) ( 0 ) ( 00) ( ) ( 0) ( 0) ( 0) ( ) ( 00) ( 0 ) ( 0) ( ) ( 0) ( ) ( ) ( ) ( 0) ( 0 ) ( 00) ( ) ( 0) ( ) ( 0) Usig Lagrage s theorem i is a subgrou the H = = H : H ad we obtai H {8 } Aalyzig the table we ca 8 seciy the subgrous o : () oe subgrou o order : H {(00)}; = () three subgrous o order : H = {(00)(0 )} = < (0 ) > H = {(00) (0)} = < (0) > ad () () H = {(00)( )} =< ( ) > () our subgrous o order : H = {(00)(0)} = < (0) > () H = {(00)( )} = < ( ) > H = {(00) (0)} = < (0) > ad H () = {(00)( )} = < ( ) > () () H8 oe subgrou o order 8: = Because is a commutative grou all its subgrous are ormal subgrous Now let s reset the set Aut ( ) o all automorhisms o From the above results we have Aut ( ) 8 where ay isomorhism / = : have the roerties ( 0) = ( a b ) ad ( 0 ) = ( c d ) ( a b ) ( c d ) a ad a / b c ad c / ad ( ) * d wwwjosaro Mathematics Sectio

7 Some alicatios rom Georgiaa Velicu Elea Catalia Gridei 6 We have: / = {ˆ x / x } xˆ = { y / x y } ad i x = k ad y = k where k k the coditio x y is equivalet to k k k k (mod ) ad the k k {0 } I coclusio we obtai x ˆ = {0ˆˆ } x the / = {0ˆˆ } Fially we have: ( a b ) {(0)( )(0)( )} ad ( c d ) {(0 )( )} ad we reset all eight automorhisms o i the table () o course how they are deied o geerators ad i table () how they actio o the elemets o i i ( 0 i ) ( 0 ) ( 0) ( 0 ) ( 0) ( ) ) ( ( 0 ) ( ) ( ) 5 0) ( ( 0 ) 6 0) ( ( ) 7 ) ( ( 0 ) 8 ) ( ( ) ( 00) ( 00) ( 00) ( 00) ( 00) ( 00) ( 00) ( 00) ( 00) ( 0 ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0) ( 0) ( 0) ( ) ( ) ( 0) ( 0) ( ) ( ) ( ) ( ) ( ) ( 0) ( 0) ( ) ( ) ( 0) ( 0) ( 0) ( 0) ( 0) ( 0) ( 0) ( 0) ( 0) ( 0) ( 0) ( ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( 0) ( 0) ( 0) ( ) ( ) ( 0) ( 0) ( ) ( ) ( ) ( ) ( ) ( 0) ( 0) ( ) ( ) ( 0) ( 0) ISSN: Mathematics Sectio

8 6 Some alicatios rom Georgiaa Velicu Elea Catalia Gridei REFERENCES [] JRDurbi Moder Algebra EdJoh Wiley New York 985 [] DBuseag FChirtes DPiciu Probleme de algebra EdUiversitaria Craiova 00 [] CNastasescu CNita CVraciu Bazele algebrei EdAcademiei 986 [] htt://mathberkeleyedu/rograms/graduate/relim-exams/archive wwwjosaro Mathematics Sectio