The Nonabelian Tensor Squares and Homological Functors of Some 2-Engel Groups

Transcription

1 Jurnal Sains dan Matematik Vol4 No1 (01) Te Nonabelian Tensor Squares and Homological Functors of Some -Engel Groups Kuasa Dua Tensor Tak Abelan dan Fungtor Homologi beberapa kumpulan Engel- Hazzira Izzati Mat Hassim 1, Nor Haniza Sarmin 1 and Nor Muainia Mod Ali 1 Department of Matematical Sciences, Faculty of Science Universiti Teknologi Malaysia, 8110 UTM Joor Baru, Joor Ibnu Sina Institute for Fundamental Science Studies Universiti Teknologi Malaysia, 8110 UTM Joor Baru, Joor 1 azzira_assim@yaoocom, 1 ns@utmmy & normuainia@utmmy Abstract Originated in omotopy teory, te nonabelian tensor square is a special case of te nonabelian tensor product Te nonabelian tensor square of a group G, denoted as G7G is generated by te symbols g7, for all g, dg subject to te relations gg'7 = ( g g'7 g ) (g7) and g7' = (g7)( g7 '), for all g, g',, 'dg were te action is taken to be conjugation Te omological functors of a group including J(G), d(g), te exterior square, te Scur multiplier, d(g), te symmetric square and J (G) d(g) are closely related to te nonabelian tensor square of te group In tis paper, te nonabelian tensor squares and omological functors of some -Engel groups will be presented Keywords nonabelian tensor square, omological functors, -Engel group Abstrak Berasal daripada teori omotopi, kuasa dua tensor tak abelan merupakan kes istimewa bagi asil darab tensor tak abelan Kuasa dua tensor tak abelan bagi suatu kumpulan G, dilabel sebagai G7G adala dijanakan ole simbol g7, bagi semua g, dg tertakluk kepada ubungan gg'7 = ( g g'7 g )(g7) dan g7' = (g7)( g7 '), bagi semua g, g',, 'dg dengan tindakannya adala kekonjugatan Fungtor omologi bagi suatu kumpulan termasuk J(G), d(g), kuasa dua peluaran, pekali Scur, d(g), kuasa dua simetrik dan J (G) d(g) adala berkait rapat dengan kuasa dua tensor tak abelan bagi kumpulan tersebut Dalam artikel ini, kuasa dua tensor tak abelan dan fungtor omologi bagi beberapa kumpulan Engel- akan ditunjukkan Kata kunci kuasa dua tensor tak abelan, fungtor omologi, kumpulan Engel- Introduction Te nonabelian tensor square of a group was originated in algebraic K-teory as well as in omotopy teory For any group G, its nonabelian tensor square, denoted as G7G is generated by te symbols g7, for all g, dg subject to te relations gg'7 = ( g g'7 g ) (g7) and g7' = (g7)( g7 ') for all g, g',, 'dg were g g' = gg'g -1 Note tat te multiplication of two elements in te nonabelian tensor square of a group cannot be obtained as a usual multiplication of two elements in a group

2 54 Jurnal Sains dan Matematik Vol4 No1 (01) 5-59 Te omological functors of a group including J(G), d(g), te exterior square, te Scur multiplier, d(g), te symmetric square and J (G) are closely related to te nonabelian tensor square of te group Te group G acts naturally on te tensor products by g (g'7) = g g'7 g and tere exists a omomorpism mapping κ, were κ : G7G' defined by κ(g7) = [g,], and [g,] = gg -1-1 Tus J(G) is te kernel of κ and is a G-trivial subgroup of G7G contained in its center Te next omological functors d(g) and d(g) denote te subgroup of J(G) generated by te elements x7x for xdg and te subgroup of J(G) generated by G G te elements (x7y) (y7x) for x, y dg respectively Te quotient group ( G ) is known as te exterior square of te group G and is denoted as G G Meanwile te symmetric square of G is defined as G G = G G ( G ) and te Scur multiplier of G is defined as M( G JG ) = ( G ) Furtermore, J JG (G) is defined as te factor group ( G ) Te study on te nonabelian tensor squares for various groups as been conducted by many researcers trougout te years Brown et al (1987) did explicit computation of te nonabelian tensor squares of all nonabelian groups up to order 0 A combination of reduction steps as been used to simplify te presentation resulting from te definition of te nonabelian tensor square Tietze transformation is performed using a computer program to assist in simplifying te presentation Ten, te simplified presentation is examined in order to determine te isomorpism type of te nonabelian tensor square Te nonabelian tensor squares of te -generator p-group of class, were p is an odd prime ave been determined by Bacon and Kappe (199) By considering te case for p =, Kappe et al (1999) extended te researc by Bacon and Kappe (199) to determine te nonabelian tensor squares of -generator -groups of class Ten, Sarmin (00) classified all infinite -generator groups of class two and determined teir nonabelian tensor squares Te omological functors were also originated in omotopy teorytis is te study of omotopy groups wic are used in algebraic topology to classify topological spaces Using teir nonabelian tensor squares, Bacon and Kappe (00) determined various omological functors for te -generator groups p -groups of class two A researc on te omological functors of infinite nonabelian -generators groups of nilpotency class two as been conducted by Mod Ali et al (007) Te computations were done using te classification of te groups and teir nonabelian tensor squares In tis researc, Groups, Algoritm and Programming (GAP) algoritms are constructed for te computation of te omological functors of tese groups Besides, Beurle and Kappe (008) classified all infinite metacyclic groups and determined teir nonabelian tensor squares Te omological functors suc as te exterior square and te Scur multiplier ave also been computed in teir researc Ramacandran et al (008) sowed te and computations of te nonabelian tensor square and omological functors of te symmetric group of order six Using te definition of te nonabelian tensor square of a group, te Cayley table of S S was found Hence, based on te nonabelian tensor square of tis groups, te omological functors were ten computed Ten, te results found were verified using GAP In tis paper, te nonabelian tensor squares and some omological functors are computed for te -Engel groups of order eigt A group G is called a -Engel group if [ g, ] = g,, = efor all elements g and in G All -Engel groups of order at most [ ] twenty ave been found by Sarmin and Yusof (006) Based on teir researc, two groups ave been identified as te -Engel groups of order eigt Tey are te Diedral group of

3 Jurnal Sains dan Matematik Vol4 No1 (01) order eigt, D 4 and te quaternion group of order eigt, Q Preliminaries Some preparatory results tat are vital in te computation of te nonabelian tensor squares and omological functors of te Diedral group of order eigt and quaternion group of order eigt are included in tis section Teorem 1(Kappe et al, 1999) Let G be a finite nonabelian -generator -group of class If ( ) [ ] [ ] [ ] G = c a b were ab, = c, ac, = bc, = 1, a= α, b= β, c= γ wit α = β = γ, ten γ γ+ 1 γ 1 ( )( ) ( ) ( ) ( ) ( ) G G = a a b b a b b a a b a b a c a b b c = C C C Teorem (Kappe et al, 1999) Let G be a finite nonabelian -generator -group of class If G ( c a ) 1 1 were a= b= γ+, [ ab, ] = γ, c= γ, [ ab, ] = ac, b, 4 [ cb] a c a γ b, =, =, γ N, ten γ 4 C C4 for γ = 1, G G C γ C γ+ 1 C γ 1 for γ Proposition 1(Brown et al, 1987) Let m= 4 r+ k, were 0 k = or Ten ( ) m J Q is isomorpic to C 4 C C, generated by ( x x)( x y) ( y y), y y, and ( x y)( y x), Lemma (Dummit and Foote, 004) Let A and B be groups wit M Lemma (Dummit and Foote, 004) k A and N B m k + respectively A B A B M N Ten ( ) ( M N) Let A= a and B = a be groups If a C, ten te quotient of cyclic group is

4 56 Jurnal Sains dan Matematik Vol4 No1 (01) 5-59 cyclic, ie A C B If a C, k ten ( k) a k = Furtermore, gcd, ( k) k t = Tus, A a = Ckt= C ( k) gcd, B a gcd, a C t were Proposition 4 (Bacon and Kappe, 00) Given a group G of nilpotency class and a generating set X for G, we ave G G = u v,u [ v,w] u,v,w X, (1) ( G) = u u, ( u v)( v u) u,v X, () ( ) ( ) ( )( ) G = u u, u v v u u,v X () Te Computation of te Nonabelian Tensor Squares and Homological Functors of D 4 and Q In tis section, te computation of te nonabelian tensor square and some omological functors of te Diedral group of order eigt, D 4 and quaternion group of order eigt, Q are sown Te results are given in te following teorems Teorem Let G = D4 wit te presentation 4 a, b: a = b = e, ba = a b Ten G G = a a b b ( a b)( b a) a b = C C4, (4) ( G) = a a b b ( a b)( b a) = C, (5) G G = a b = C 4, (6) G = a a b b a b b a = C (7) G G = a a b b a b = C C (8) 4 Proof Based on Teorem 1, by letting α = β = γ = 1, te nonabelian tensor square of G is determined as follows: ( )( ) G G = a a b b a b b a a b C C C C 4

5 Jurnal Sains dan Matematik Vol4 No1 (01) Tis sows tat D4 D4 is isomorpic to tree copies of cyclic group of order two and a copy of cyclic group of order four Note tat a a = b b = ( a b)( b a) = and a b = 4 By () in Proposition 4 and order restriction on te generators, it follows tat ( ) ( )( ) G = a a b b a b b a C Tus (5) olds Using Lemma, Lemma and observing tat ( G) ( G G), G G = ( G G) ( G) ( ) ( ) ( a b)( b a) a a b b a b = a a b b a b b a = a b G = a b ( )( ) 1 Ten, by te order restrictions on te generators, G G C, 4 te desired result Similarly as proof of (5), using () in Proposition 4, we ave ( G) ( a a) ( a b)( b a) ( b b) = From Lemma, ( a a) ( b b) = = = 1 By tese and te order restrictions gcd(, ) G C Tus, (7) olds on te generators, we obtain ( ) Note tat ( ) a a is a proper subgroup of index two in a a and ( b b) is a a a b b proper subgroup of index two in b b Tus, ( a a) ( b b) C Togeter wit Lemma, Lemma and observing tat ( G ) ( G G ), we obtain G G = ( G G) ( G) a a b b = a b ( a a) ( b b) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( a b)( b a) ( a b)( b a) = a a G b b G a b G = a a b b a b Using te order restrictions on te generators, tis leads to te desired result (8)

6 58 Teorem 4 Let G = Q wit te presentation Jurnal Sains dan Matematik Vol4 No1 (01) a, b: a = b = e, a = b, ba = a b Ten G G C C ( ) 4 4, (9) J G C C (10) Proof Since te quaternion group of order eigt is a finite nonabelian -generator -group of class were G ( c a ) b, wit γ γ 4 [ cb, ] a c, a b a γ + 1 = b =, [ ab] γ, =, c γ 1 =, [ ab] =, ac, = = for γ = 1, ten based on Teorem, G G C C 4 Tis sows tat Q Q is isomorpic to two copies of cyclic group of order two and two copies of cyclic group of order four From Proposition 1, since for Q, m = and k =, ten ( ) ( )( ) ( ) ( )( ) J G = x x x y y y y y x y y x C C C C C Tus, (10) olds Conclusion In tis paper, te nonabelian tensor squares and omological functors ave been computed for all -Engel groups of order eigt Acknowledgement Hazzira Izzati Mat Hassim would like to tank te Ministry of Higer Education for er MyPD scolarsip References Bacon, M R, & Kappe, L-C (199) Te nonabelian tensor square of a -generator p-groups of class Arc Mat, 61, Bacon, M R, & Kappe, L-C (00) On capable p-group of nilpotency class twoillinois Journal of Mat, 47, 49-6 Beurle, J R, & Kappe, L -C (008) Infinite metacyclic groups and teir nonabelian tensor squares of free nilpotent groups of finite rank Contemporary Matematics, 470, 7 44 Brown, R & Loday, J-L (1987) Van Kampen Teorems for diagrams of spaces Topology 6, 11-5 Brown, R, Jonson, D L, & Robertson, E F (1987) Some computations of nonabelian tensor products of groups Journal of Algebra, 111, Dummit, D S, & Foote, R M (004) Abstract Algebra Jon Wiley and Son, San Francisco, USA Kappe, L -C, Visscer, M P, & Sarmin, N H (1999) Two-generator-two-groups of class two and teir nonabelian tensor squaresglasgow Mat J, 41,

7 Jurnal Sains dan Matematik Vol4 No1 (01) Mod Ali, N M, Sarmin, N H, & Kappe, L -C (007) Some omological functors of infinite nonabelian -generators groups of nilpotency class Proceeding of 15 t Matematical Sciences National Conference (SKSM 15), Ramacandran V A R, Sarmin, N H, & Mod Ali, N M (008) Te nonabelian tensor square and omological functors of S Proceeding of Regional Annual Fundamental Science Seminar Sarmin, N H (00) Infinite two-generator groups of class two and teir nonabelian tensor squares International Journal of Matematics and Matematical Sciences, (10), Sarmin, N H, & Yusof, Y (006) On -Engel Groups of Order at Most 0 Tecnical Report of Department of Matematics, Universiti Teknologi Malaysia, LT/M No5