Some Applications of Metacyclic p-groups of Nilpotency Class Two

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1 Research Journal of Alied Sciences, Engineering and Technology 4(3): , 01 ISSN: Maxwell Scientific Organization, Submitted: Aril 5, 01 Acceted: May 13, 01 Published: December 01, 01 Some Alications of Metacyclic -Grous of Nilotency Class Two 1 Abdulqader Mohammed Abdullah Bin Basri, Nor Haniza Sarmin, Nor Muhainiah Mohd Ali and 3 James R. Beuerle 1 Deartment of Mathematical Sciences, Faculty of Science, UniversitiTeknologi Malaysia, UTM Johor Bahru, Johor, Malaysia Deartment of Mathematical Sciences, Faculty of Science and Ibnu Sina Institute For Fundamental Science Studies, Universiti Teknologi Malaysia, UTM Johor Bahru, Johor, Malaysia 3 Mathematics and Statistics Deartment Elon University, NC 744, USA Abstract: A grou G is metacyclic if it contains a cyclic normal subgrou K such that G/K is also cyclic. Metacyclic -grous classified by different authors. King classified metacyclic -grous. Beuerle (005) classified all finite metacyclic -grous. A grou is called caable if it is a central factor grou. The urose of this study is to comute the eicenter of finite nonabelian metacyclic -grous of nilotency class two, for some small order grous, using Grous, Algorithms and Programming (GAP) software. We also determine which of these grous are caable. Keywords: -grous, metacyclic grous, eicenter and caable grous INTRODUCTION A finite -grou G is called metacyclic if it has a cyclic normal subgrou K such that G/K is also cyclic. Finite metacyclic grous can be resented with two generators and three defining relations. Much attention has been given to some secific tyes of metacyclic grous. Zassenhaus discussed metacyclic grous with cyclic commutator quotient, as did Hall. Metacyclic - grous of odd order have been classified by (Beyl et al., 1979). Significant rogress was made when Sim (1994) roved that every finite metacyclic grou can be decomosed naturally as a semidirect roduct of two Hall subgrous (Hemel, 000). For a grou G the eicenter of G is defined as follows: Definition 1: Beyl et al. (1979) the eicenter, Z * (G) of a grou G is defined as: 1{NZ(E); (E, N) is a central extension of G} It can be easily seen that the eicenter is a characteristic subgrou of G contained in its center. The notion of the caability of a grou was first studied by Baer (1938) who initiated a systemic investigation of the question which conditions a grou G must fulfill in order to be the grou of inner automorhisms of some grou H, i.e., G H ZH The term was coined by Hall and Senior (1964) who defined a caable grou as equal to its central factor grou. The caability for some grous has been studied by many authors including Baer (1938) who characterized finitely generated abelian grous which are caable in the following theorem: Theorem 1: Baer (1938) Let A be a finitely generated abelian grou written as A= Z Z Z such that n n K 1 n k each integer n i+1 is divisible by n i and Z 0 = Z, the infinite cyclic grou. Then A is caable if and only if k$ and n k!1 = n k. Beyl et al. (1979) established a necessary and sufficient condition for a grou to be a central quotient in terms of the eicenter as follows: Theorem : Beyl et al. (1979) A grou G is caable if and only if Z * (G) = 1. With the hel of this characterization, Beyl et al. (1979) were able to determine the extra secial -grous which are caable and describe the caable finite metacyclic grous. Beuerle (005) gave a resentation of metacyclic - grous in the form: Corresonding Author: Nor Haniza Sarmin, Deartment of Mathematical Sciences, Faculty of Science and Ibnu Sina Institute For Fundamental Science Studies, Universiti Teknologi Malaysia, UTM Johor Bahru, Johor, Malaysia 517

2 Res. J. Al. Sci. Eng. Technol., 4(3): , 01 m m k 1 r G = a, b; a = 1, b = a, bab = a where, mn r k m m, 0, > 0,,, k( r 1) m n and r 1 Metacyclic -grous are divided by Beuerle (005) into two categories as follows: ( αβεδ ) G,,,, ± = a, b; a = 1, b = a, bab = a where, r = "!* +1 or r = "!*!1 α β α ε 1 r We say that the grou is of ositive or negative tye if r = "!* +1 or r = "!*!1, resectively. For short, we write G + and G! for G(", $, g, *, +) and G(", $, g, *,!), resectively. The following theorem, due to King (1973), gives a standardized arametric resentation of metacyclic - grous. Theorem 3: Let G be a finite metacyclic -grou. Then there exist integers ", $, *, g with ", $>0 and *, g nonnegative, where *#min {"!1, $} and *+g#" such that for an odd rime, G (", $, g, *, +). If =, then in addition "!*>1 and G (", $, g, *, +) or G (", $, g, *,!), where in the second case g#1. Moreover, if G is of ositive tye, then *>0 for all. The next roosition lists some of the basic roerties and various subgrous of metacyclic -grous: Proosition 1: Beuerle (005) Let G be a grou of tye G (", $, g, *, ±) and k$1. Then we have the following results for the order of G the center of G, the order of the center of G the derived subgrous of G the order of the derived subgrous of G and the exonent of G: G + G! G ± "+$ "+$ δ Z(G ± ) a, b δ α 1 max { 1, δ } a, b Z(G ± ) "+$!* 1+$!max{1, *} ( k+1 (G ± ) a k α δ a k ( k+1 (G ± ) max{0, k*!(k!1)"} max{0, "!k} Ex(G ± ) max{", $+g} max{", $+g} Moreover, if $>max{1, *} then Z (G! ) is cyclic if and only if g = 1. The nonabelian tensor square is a secial case of the nonabelian tensor roduct which has its origin in K-theory as well as homotoy theory. It is defined as follows: Definition : Brown et al. (1987) for a grou G, the nonabelian tensor square, GqG is generated by the symbols gqh, where g, h0g, subject to the relations: ' ` gg h ( g ' g = g h)( g h) ' and g hh ( g h)( h h ' = g h) for all g, g!, h, h!,g, where h g = hgh!1 denotes the conjugate of g by h. A factor grou of the tensor square is the exterior square, defined as follows: Definition 3: Brown et al. (1987) for any grou G the exterior square of G is defined as: ( ) G G= G G G where, L(G) = +xqx: x0g, and L(G) is a central subgrou of GqG. In (Ellis, 1995) the tensor center of a grou is defined as: = { : = 1 ; } Z G a G a g g G Similar to the tensor center, the exterior center can be defined as: = { : = 1 ; } Z G a G a g g G Here 1 q and 1 v denote the identities in GqG and GvG resectively. It can be easily shown that Z q (G) and Z v (G) are characteristic and central subgrous of G with Z q (G)fZ v (G). Though the criterion for caability stated in Theorem is easily formulated, the imlementation is another matter. As in all cases before, this still requires the cumbersome rocess of evaluating the factor grous. Ellis (1998) connects the eicenter with the exterior center, roviding a much more efficient rocedure to determine caability once the nonabelian tensor square is known and obtains the desired external characterization of the eicenter as follows: Theorem 4: Ellis (1998) For any grou G, the eicenter coincides with the exterior center, i.e.: Z*(G) = Z^(G) 518

3 Res. J. Al. Sci. Eng. Technol., 4(3): , 01 The method of using tensor squares in the determination of caability was imlemented by several authors. Beuerle and Kae (000) characterized infinite metacyclic grous which are caable. Beuerle (005) classified all finite nonabelian metacyclic -grous. Using this classification, we comute the eicenter of finite nonabelian metacyclic - grous of nilotency class at least three, for some small order grous, by using Grous, Algorithms and Programming (GAP) software. By using the criterion stated in Theorem, we determine which of them are caable. Grous, Algorithms and Programming (GAP) software is used as a tool to verify the results found in determining the caability of the grous studied. GAP is a system for comutational discrete algebra, with emhasis on comutational grou theory. GAP rovides a rogramming language, a library of functions that imlement algebraic algorithms written in the GAP language as well as libraries of algebraic objects such as for all non-isomorhic grous u to order 000 (Rainbolt and Gallian, 010). Classification of metacyclic -grous of nilotency class two: In this section, the classification of all finite nonabelian metacyclic -grous of nilotency class two, done by Beuerle is stated. Theorem 5: (Beuerle, 005) Let G be a nonabelian metacyclic -grou of nilotency class two. Then G is isomorhic to exactly one grou in the following list: 1., ; 1, [, ] α β α δ G a b a = b = a b = a where, ", $, *,N, "$*, *#$. G Q = +a, b, a 4 = 1, b = [a, b] = a, the grou of quaternion of order 8. EPICENTER COMPUTATIONS In this section, we use GAP to develo aroriate coding for comuting the eicenter for some small order grous of tye (1), where is odd and tye (). We rovide GAP rogrammes to generate general codes and examles of grous of tye (1) and tye (). Tye (1): First we use GAP rogramme to generate all finite nonabelian metacyclic -grous of tye (1) and construct some examles. Generating tye (1): GAP coding to generate all finite nonabelian metacyclic -grous of tye (1) is develoed as follows: tye1: = function (, l, beta, n) local F, m, a, b, G; for m in [1..beta] do if n<=l and m>=n and n>=1 then F: = FreeGrou (); a: = F.1; b: = F. G: = F/[a^(^l), b^(^m), (a*b*a^(-1)*b^(-1))^(- 1)*a^(^(l-n))]; Print("=",," alha=",l," beta=",m," delta =",n," Order(G):",Order(G)," Order of Eicenter (G):",Order (Eicenter (G)),"\n"); fi; od; end; We define the grou with four arameters which are, ", $ and *. The GAP code below is used to create the resentation of the grou. F: = FreeGrou (); a: = F.1; b: = F. G: = F/[a^(^l), b^(^m), (a*b*a^(-1)*b^(-1))^(- 1)*a^(^(l-n))]; Constructing examles of tye (1): Now, we define the arameters and ut them in GAP. For examle, we define = 3, " =, $ = 1,..., 6 and * = 1. First we need to read the file and call the function. ga> Read("metacyclic_tye3.g"); ga> tye3 (3, 3, 4, ); = 3 alha = beta = 1 delta = 1 Order(G):7 Order of Eicenter(G):3 = 3 alha = beta = delta = 1 Order(G):81 Order of Eicenter(G):1 = 3 alha = beta = 3 delta = 1 Order(G):43 Order of Eicenter(G):3 = 3 alha = beta = 4 delta = 1 Order(G):79 Order of = 3 alha = beta = 5 delta = 1 Order(G):187 Order of Eicenter(G):7 = 3 alha = beta = 6 delta = 1 Order(G):6561 Order of Eicenter(G):81 Using the coding as before, we got the following results. ga> tye1 (3, 3, 6, 1); = 3 alha = 3 beta = 1 delta = 1 Order(G):81 Order of = 3 alha = 3 beta = delta = 1 Order(G):43 Order of Eicenter(G):3 = 3 alha = 3 beta = 3 delta = 1 Order(G):79 Order of Eicenter(G):1 ga> tye1 (3, 4, 5, ); = 3 alha = 4 beta = delta = Order(G):79 Order of 519

4 Res. J. Al. Sci. Eng. Technol., 4(3): , 01 = 3 alha = 4 beta = 3 delta = Order(G):187 Order of Eicenter(G):3 ga> tye1 (5,, 5, 1); = 5 alha = beta = 1 delta = 1 Order(G):15 Order of Eicenter(G):5 = 5 alha = beta = delta = 1 Order(G):65 Order of Eicenter(G):1 ga> tye1 (7,, 5, 1); = 7 alha = beta = 1 delta = 1 Order(G):343 Order of Eicenter(G):7 ga> tye1 (9,, 5, 1); = 9 alha = beta = 1 delta = 1 Order(G):79 Order of ga> tye1 (11,, 5, 1); = 11 alha = beta = 1 delta = 1 Order(G):1331 Order of Eicenter(G):11 ga> tye1 (13,, 5, 1); = 13 alha = beta = 1 delta = 1 Order(G):197 Order of Eicenter(G):13 ga> LogTo(); ga> quit; RESULTS ANALYSIS Summary of results for tye (1): For short, we write E (G) for the eicenter of G. A summary of GAP results for tye (1) is given in Table 1. Remark 1: Table 1 shows that the order of all grous comuted satisfies "+$ as stated in Proosition 1. Tye (): With similar objective as for grous of tye (1), we roduce GAP algorithms for grous of tye () and comute the order of the eicenter as follows: ga> F: = FreeGrou (); <free grou on the generators [ f1, f ]> ga> a: = F.1; b: = F.; f1 f ga> Q: = F/[a^4,b^*(a*b*a^(-1)*b^(-1))^(-1), b*a^(- 1)*b^(-1)*a^(-1), b^*a^(-)]; <f grou on the generators [ f1, f ]> ga> Order (Eicenter(Q)); ga> LogTo(); Caability determination: Table 1 shows that the grous of tye (1) have trivial eicenter with " = $. GAP results shows that the grou of tye (), the quaternion grou of order 8, has nontrivial eicenter. Therefore, by Theorem, we conclude that grous of tye (1) are caable with Table 1: GAP results for tye (1) # " $ * G E (G) " = $ and the rest are not caable. Similarly, we conclude that the quaternion grou of order 8 is not caable. It is worth to mention that Bacon and Kae (003) showed the same results for tye (1), where is odd. Also, the results for tye () consistent with the results gotten by Ellis (1998) who showed that the quaternion grou: [ ] Q = aba, ; = 1, b = ab, = a n n of order n is not caable. Based on the above, we write the results in the following theorem: Theorem 6: Let G be a metacyclic -grou of nilotecny class two. Then G is caable if and only if G is of tye (1) with " = $. CONCLUSION In this study we have comuted the eicenter of finite nonabelian metacyclic -grous of nilotency class two, for some small order grous and determined their caability. We showed that the eicenter of grous of tye (1) is trivial with " = $ and the eicenter of the grou of tye (), the quaternion grou of order 8, is nontrivial. Also, we showed that the grous of tye (1) are caable with " = $ whereas the quaternion grou of order 8 is not caable. ACKNOWLEDGMENT The authors would like to acknowledge Universiti Teknologi Malaysia for the artial funding of this research through the Research University Grant (RUG) vot No 00H48. 50

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