Groupoid and Topological Quotient Group

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1 lobal Jounal of Pue and Appled Mathematcs SSN Volume 3 Numbe 7 07 pp Reseach nda Publcatons oupod and Topolocal Quotent oup Mohammad Qasm Manna Depatment of Mathematcs Collee of Scence Al-Qassm Unvesty P Box: 6644-Buadah: 545 Saud Aaba Abstact n ths pape we nduced some esults whch ae make connecton between the concept of oupod and the quotent oup Futhe we ntoduce some new popetes to the dea of oupod n quotent oup quotent oup-oupod Moeove we ve defntons to topolocal quotent oup-oupod and poduct of quotent oup-oupods and dscuss some possessons to these defntons Keywods and phases: Topolocal oupod oup-oupod topolocal oup Mathematcal Subject Classfcaton 00:A 6BXX 54 0 ntoducton ndeed Mucuk [8] ntoduced the defnton of quotent oup-oupod fom the fact that; the oup-oupod s equvalent to cossed module Actually we need moe explan to ths defnton to exploe and constucton some new popetes and notons based to ths defnton As n oup theoy we dscuss the elatonshp between the oup-oupod and nomal suboup-oupod thouh the concept of quotent oup to nduce some esults whch ae elated to new fom to the defnton of oupods n quotent oup quotent oup-oupod and study some new popetes to the dea of ths defnton

2 374 Mohammad Qasm Manna Also n ths atcle we ntoduce a defnton to topolocal quotent oup-oupod and nduced some popetes of ths defnton ove the noton of topolocal quotent oup theoy and constuct the fom of poduct quotent oup-oupods Pelmnaes A oupod [] s small cateoy conssts of two sets and called espectvely the set of elements o aows and the set of objects o vetces of the oupod toethe wth two maps : called espectvely the souce and taet maps the map : wtten as x= x whee x s called the dentty element at x n and s called the object map and the patal multplcaton map : wtten h h on the set h : h These tems must satsfy the followns axoms: h = h and h = h k= h k 3 x x x 4 = and = fo all h k and x Fo a oupod we wll denote to the nvese map by; 5 : such that A mophsm of oupods and s a functo that s t conssts of a pa of functons f : and f : such that; f ab f a f b f a f a 3 f f f f 4 f Whee a b s defned f A suboupod of s a subcateoy of whch s also a oupod We say that y s full wde suboupod f fo all x y then x y x

3 oupod and Topolocal Quotent oup 375 a Moeove s called nomal suboupod f wde and a x a y o y a x fo any x y and y a x oup-oupod n fact the noton of oup-oupod s equvalent to many concepts n cateoy theoy and we pesent these concepts as follows the coss modules ntenal cateoy n oup and the oup object n cateoy of oupods see [9] The follown defnton and popetes ntoduced by [3] Defnton A oup-oupod s a oupod endowed wth a oup stuctue such that the follown maps whch ae called espectvely addton nvese and unt ae mophsm of oupods: m : h h oup multplcaton : u oup nvese map 3 e: whee s snleton So by 3 f the dentty fo the oup stuctue on s e then e s the dentty fo the oup stuctue on the aows of n oup-oupod fo h the oupod composte s denoted by h when h and the oup addton by h A mophsm of oup-oupods f : s a mophsm of the undelyn oupods pesevn the oup stuctue The follown Poposton a pea n Bown and Spence [3] Poposton Let be a oup-oupod f y z b a b a b a b a y y e y e y y a a a a x a y b and a x y then Defnton 3[7 Defnton ] Let be a oup-oupod and Then s a suboup-oupod of f fom a oup-oupod Futhe s wde f and full f x y x y fo all x y Moeove f s a suboup-oupod and the set of aows of s nomal suboup

4 376 Mohammad Qasm Manna then the set of objects s nomal suboup [7 Poposton 5] So s called nomal suboup-oupod see also [8 Defnton 30] The follown defnton appeas n [5] Defnton 4 A topolocal oup-oupod s a oup-oupod wth a topoloes on and such that all the mophsms of oup-oupod ae contnuous that s: m u and A mophsm of topolocal oup- oupods f : s contnuous 3 Quotent oup-oupod n ths secton we study the popetes of quotent oup on oup-oupod to defne the concept of quotent oup-oupod Let be a oup and be nomal suboup of The oup s called the quotent oup and the map [4] q: such that s called quotent map Poposton 3 Let be a oup-oupod and be a nomal suboup-oupod of then s a suboup of and s a nomal suboup of Poof Let h t Snce and u ae mophsms of oups then h h t t mples that h t then h t Smlaly we can show that s a suboup of To pove that s a nomal suboup of Let and h h then we have that h h and ths mples that h h h h

5 oupod and Topolocal Quotent oup 377 Fom above esult we defne the set : wth the multplcaton on t as So t s clea that s a oup quotent oup Futhe ths leads to defne a quotent map : ts easy to see that s well-defned onto and homomophsm Poposton 3 Let be a oup-oupod and be a nomal suboup-oupod then Poof Defne h : h ts easy to see that h s onto and homomophsm of oups Futhe we have h Ke h : : So t s obvous that Poposton 33 Let be a oup-oupod and be a nomal suboup-oupod of then : s a suboup of Poof Clea Poposton 34 Let be a oup-oupod and be a nomal suboup-oupod of then ae somophc oups Poof Defne : then s welldefned homomophsm and onto and : Ke : = Then we have the follown somophc Remak 35 Fom above esult we can defne the somophsm oups as

6 378 Mohammad Qasm Manna : also we have : Poposton 36 Let be a oup-oupod and be a nomal suboup-oupod of then : such that s onto mophsm of oups Poof Well-defned; Suppose that then we have Snce s a nomal suboup-oupod then and mples that = omomophsm; whee nto; Let snce Then thee s n such that n the follown poposton we defne the set as; h h h h : Poposton 37Let be a oup-oupod and nomal suboup-oupod then then Poof Suppose that h h h h and snce the souce and taet maps ae mophsm of oups then we have h and then h mples that h h mples that h h h h

7 oupod and Topolocal Quotent oup 379 Convesely suppose that h Snce h h h h h Then we have Fom and we have Poposton 38 Let be a oup-oupod and be a nomal suboup-oupod of then : such that s onto mophsm of oups Poof Fom Poposton 37 we note that s well-defned omomphsm; = = nto; Suppose that then snce then thee s n such that n the follown we ntoduce a defnton to quotent oup-oupod smla to [8 Defnton 38] at most Defnton 39 Let be a oup-oupod and be a nomal suboup-oupod of Then and the follown maps called quotent oup-oupods The souce map : The taet map :

8 380 Mohammad Qasm Manna 3 The object map : x 4 The multplcaton map : 5 The nveson map oupod : 6 The multplcaton oup m : m x 7 The nvese map oup u : u Remak 30 We note that fom a bove defnton All maps ae well-defned The quotent map q: ae mophsm of oupods 3 Fom 4 Suppose that then we have whch mples that Also fom the defnton of souce and taet maps n quotent oup-oupod we have espectvely and So fom and we have We note that fom Poposton 34 Popostons and Poposton 38 that Remak 3 Fom Defnton 39 we have the follown llustaton; Snce q : and q : ae quotent maps then fom the follown commutatve dam q q We have and x x Futhe the nveson map fom

9 oupod and Topolocal Quotent oup 38 q q Such that Also the constuctons of multplcaton map come fom the follown dam q See Remak 35 Poposton 3 Let be quotent oup-oupod then Poof Clea Remak 33 Thee s anothe way to pove the Poposton 37 Want to pove that Snce n the othe hand So we have

10 38 Mohammad Qasm Manna Poposton 34 Let be a quotent oup-oupod Then ts satsfes the ntechane law Poof Let b a b a a b a b a = b a = ba a a b whee ba and ae defned then Poposton 35 Let be a quotent oup-oupod Then the follown m : such that u : such that ae mophsms of oupods Poof bseve that m a b m a b = a b a b = a b a b = a b a b m a m b Also we have m a b a b ab m a b Futhe m m and m m m m Smla above u a b u a u b Also u a u a Moeove u u and u u u u Poposton 36 Let ae mophsm of oups be a quotent oup-oupod Then and

11 oupod and Topolocal Quotent oup 383 Poof Fst of all we have n the othe hand Then Also snce Then we have So s a mophsm of oupods and smla we can show that s mophsm of oups Moeove t s easy to see that and Also x y x and x xy y x x x Theefoe the nveson and object maps ae mophsms of oupods Example 37 Let be a oup bseve that s a oup-oupod wth the object set and the mophms ae the pas a b The souce and taet maps ae defned by a b b and a b a the composton of oupod s defned by a b b c a c and the oup multplcaton s defned by a b d c ad bc a a a Theefoe Also the object map defned s a oup-oupod Now let be a nomal suboup n Then s nomal suboup-oupod n So s a quotent oup-oupod wth the object set and the mophms ae the pas a b The souce taet and object maps ae defned by a b a b b a b a and defned the object map a a a espectvely The composton of quotent oupoupod defned as; Futhe the oup multplcaton; a b b c a b b c a c a b d c ad bc Also nvese oup map; u a b a b Moeove the nveson map oupods defned as a b b a f : K be onto mophsm of oup-oupod then Kefs Poposton 38 Let nomal suboup-oupod whee Ke f Poof Snce Kefand s a suboupod Let Kef Kef ae nomal suboup Then we need to pove that Kef a defned then a bkefsuch that b

12 384 Mohammad Qasm Manna f a b f a f b e e e f a f a Poposton 39 Let e e f : K be onto mophsm of oup-oupods be a Kef Then thee exst a unque onto h : K such that f hq whee q s a nomal suboup-oupod such that mophsm of oup-oupods quotent map and s a quotent oup-oupod Poof Defne h f t s easy to see that h a unque mophsm of oups Futhe we need to pove that h s a mophsm of oupods Let h h f f f h h Also h f f h Futhe h K K f f h q h Smla we have h Moeove h K h K h Remak 30 Fom a bove poposton f then h wll become an somophsm of oups [4] So we wll call to a bjecton mophsm of oupoupods t an somophsm oup-oupods Kef Poposton 3 Let be a oup-oupod and and K ae nomal suboupoupod then K s a nomal suboup-oupod n whee K s nomal suboup-oupods whee K K Poof Snce K s a nomal suboup then t s enouh to pove that K s a suboupod Let a b K such that a b defned then a b hk hk hk h k h k h k h k mples that a b K Futhe t s h k clea that a h k h k K So K s nomal suboup-oupod K K

13 oupod and Topolocal Quotent oup 385 Clea Poposton 3 Let and K ae suboup-oupod n a oup-oupod whee s a nomal suboup-oupod and s a quotent oup-oupod then s nomal suboup-oupod of K K s nomal suboup-oupod of K 3 Thee s somophsm oup-oupod between K K and K Poof and clea 3Let k k K such that k k s defnd then k k k k K Also we have k k K Then K s a suboup-oupod and smla we can pove that K K s a suboupoupod of Now defne f : K K f k k Then f s onto mophsm of oups Also f s a mophsm of oupods fom the follown commutatve dam: f K K f K K Futhe we have f k k k k k k f k f k and f k k k f k Also by the second fundamental theoem of oup somophsm we have : K K K whee Ke f K an somophsm of oups and t s easy to see that s a mophsm of oupods 4 Toplocal Quotent oup-oupod n ths secton we ntoduce the defnton of toplocal quotent oup-oupod and dscuss some of ts popetes Let be a topolocal oup [] and be a nomal suboup snce the quotent map q : s contnuous open and onto then topolocal oup such that: m : and u : ae contnuous

14 386 Mohammad Qasm Manna n the follown we consde s a oup-oupod s a nomal suboup-oupod and s a quotent oup-oupod Defnton 4 A oupod n topolocal quotent oup topolocal quotent oupoupod s a quotent oup-oupod wth a topoloes on the set of aows and the set of objects such that all the mophsms of a quotent oupoupod ae contnuous that s: m u and Poposton 4 Let be a topolocal oup-oupod and be a nomal suboupoupod Then s topolocal quotent oup-oupod Poof Snce s topolocal oup Want to pove that and ae contnuous; Snce q then s contnuous fom the fact that q s q contnuous and q s a quotent map n smla way t s easy to see that ae contnuous t emans to pove that Defne and q Such that q whee q and ae quotent maps Then t s clea that s contnuous Moeove : s a contnuous and snce then we have s contnuous Example 43 Let be a topolocal oup-oupod then we have the set e e e s nomal suboup-oupod wth one object e Then s a topolocal quotent oup-oupod such that the set of aows s and the set of objects s The souce taet and object maps defnd a a a a and x x ae contnuous Moeove the nveson map a a and the multplcaton map;

15 oupod and Topolocal Quotent oup 387 : a b a b a b ae contnuous and the set of aows and the set of objects ae topolocal quotent oups Poposton 44 Let be a topolocal quotent oup-oupodthen the set of aows and the set of objects ae topolocal oups Poof Clea Defnton 45 Let f : be a contnuous mophsm of topolocal oupoupods Then f called topolocal somophsm oup-oupod f and only f ts bjecton and open Poposton 46 Let f : be onto contnuous mophsm of topolocal oupoupods Then h : Ke f s a contnuous smophsm oup-oupods such that f hq whee q s a quotent map and Ke f s a quotent oupoupodmoeove h s topolocal somophsm oup-oupods f and only f f s open map f h q Poof Snce then ts clea that h s contnuous and somophsm of oupoupods see Poposton 39 and Remak 30 Also f h s open and snce q s open and f h q then f s open Convesley suppose that f s open then snce q s quotent map ts easy to see that h s open Poposton 47 Let be a topolocal oup-oupod and be a nomal suboup-oupod such that s a quotent oup-oupodthen the aows of and the objects ae ausdoff dscete f and only f and ae closed open n and espectvely Poof Clea See [ Poposton 38] Poposton 48 Let and K ae suboup-oupod n a topolocal oup-oupod whee s a nomal suboup-oupod and let q : the quotent map Then the bjecton : K K K contnuous

16 388 Mohammad Qasm Manna n topolocal oupod f the taet o the souce maps ae open then s called nteo open topolocal oupod whch means that the othe maps n ae open maps Poposton 49 Let s openthen be a topolocal quotent oup-oupod f one of and ae open Poof Suppose that s open then and ae open maps Snce maps q and q ae quotents and q q then t s clea that s open Also snce s open and then s open Moeove snce q then we have s open see Poposton 4 and snce whee s open map then s open 5The Poduct of Quotent oup-oupods ee we defne the concept of the poduct of quotent oup-oupod Let : be a famly of topolocal oup-oupods then the poduct wth fom a topolocal oup-oupod [6 Poposton 4] n the follown paaph we ntoduce the concepts of the poduct quotent oup-oupods Let : be a famly of quotent oup-oupods whee s a nomal suboup-oupod fo all Defne the set of aows wth the set of objects such that s the set of all tuples x fo each x The set of aows s the set of all tuples fo each and the composton multplcaton and nvese defned espectvely; fo each fo each 3 u Also we defne the object oup multplcatons; 4 x y x y

17 oupod and Topolocal Quotent oup 389 fo each y x The souce the taet and the object map ae defned as x x Also the nveson map s 8 Futhe satsfes the ntechane law e t k = t k = t k = k t = k t = k t Moeove and ae mophsms of oups; = = = Also we can see that;

18 390 Mohammad Qasm Manna x y x y Poposton 5 Let : be a famly of topolocal quotent oupoupods then the poduct s topolocal quotent qoup-oupod whee s a nomal suboup-oupod fo all Poof The pevous dscusson ndcates that s a quotent oup-oupod Also ts clea that the multplcaton and nvese maps of oup n the follown; m : m u : u ae contuous maps see[ pp37-39] Futhe we have the multplcaton map of oupods; 3 : fo each / P P P / ae contnuous map fom the fact that Moeove we have to pove that and ae all contnuous maps But whee a P P b P P c P P d P P P : and P : and quotent oup-oupod ae pojecton maps Theefoe ae all contnuous maps Ths mples that s a topolocal Refeences [] Boubak N eneal Topoloy Pas: eman 966 [] Bown R Topoloy and oupods BookSue LLC Noth Caolna 006

19 oupod and Topolocal Quotent oup 39 [3] Bown R and Spence C B -oupod Cossed modules and the fundamental oupod of topolocal oup Pos Konn Ned Akad V Wet [4] unefod T W Aleba Spne Vela 980 [5] cen and Fath zcan A Topolocal cossed modules and -oupods Aleba oups eom 00 8: [6] Mann a M Q Remak on Topolocal oup-oupod PJMMS Volume 8 No [7] Mann a M Q Topolocal Suboup-oupod FJMS Volume 6 No 0 pp [8] Mucuk Sahan T and Alemda N Nomalty and Quotents n Cossed Modules and oup-oupods Stuct 05 3:45-48 [9] Pote T Extensons cossed module and ntenal cateoes n cateoes of oups wth opeatons Poc Ednb Math Soc

20 39 Mohammad Qasm Manna