In this paper, the construction of permutation groups which include symmetric groups, alternating groups, dihedral groups and the groups generated by the semidirect product (wreath products) of two permutation groups. The perfect residuum of the constructed groups has been obtained and their primitivity status has been investigated which enable us to formulate some results on such properties concerning the groups. A standard computer program, Groups algorithm and programming (GAP) has been employed in enhancing and validating the result obtained. Keywords: Primitive groups, Perfect Residuum, permutation groups

Atomsa, Y.

Bello, M.

Danjuma, Mustapha

Musa, Sani

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.7 No.24 2017  78 PRIMITIVITY OF PERFECT RESIDUUM OF PERMUTATION GROUPS  M. Bello Mathematics and computer science department, Federal University Kashere, Gombe State, Nigeria P.M.B 0182, Gombe State Nigeria. Tel: +2348060971976,   Email: mbatap560@gmail.com  Mustapha Danjuma Mathematics and Statistics Department, School of science, Abubakar Tatari Ali Polytechnic, Bauchi State Nigeria Tel: +2348037025131,   Email: mustaphdanjuma@gmail.com  Y. Atomsa Mathematics and computer science department, Federal University Kashere, Gombe State, Nigeria P.M.B 0182, Gombe State Nigeria. Tel: +2347034945570,   Email: au.nlar@gmail.com   Sani Musa  Mathematics and Statistics Department, School of science, Abubakar Tatari Ali Polytechnic, Bauchi State Nigeria Tel: +2348028909358,   Email: aksanimusa@gmail.com  The research is financed by Federal University Kashere and Abubakar Tatari Ali Polytechnic, Bauchi  Abstract In this paper, the construction of permutation groups which include symmetric groups, alternating groups, dihedral groups and the groups generated by the semidirect product (wreath products) of two permutation groups. The perfect residuum of the constructed groups has been obtained and their primitivity status has been investigated which enable us to formulate some results on such properties concerning the groups. A standard computer program, Groups algorithm and programming (GAP) has been employed in enhancing and validating the result obtained.   Keywords: Primitive groups, Perfect Residuum, permutation groups.  Introduction The concept of residuum is very important in the theory of  permutation groups more especially on investigating the solvability status of groups. The research adopted the concept of M. Bello et all (2017), work on a numerical search for polycyclic and locally nilpotent permutation groups. Definition 1.1 A smallest normal subgroup of a group that has solvable factor group is called a perfect residuum of the group. Definition 1.2 A subgroup N of a group G is normal in G if the left and right cosets are the same, that is if   and a subgroup H of G. Definition 1.3 A group G is said to act on a set X when there is a map    such that the following conditions holds for all elements . i.  where e is the identity element of G ii.  Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.7 No.24 2017  79 Definition 1.4 A group action is transitive if it possess only a single group orbit. That is for every pair of elements x and y, there is a group element g . A group is said to be intransitive if it is not transitive. If for every two pairs of points    and  there is a group element  , then the group action is called doubly transitive. Similarly, a group action can be triply transitive and in general, a group action is k-transitive if every set   of 2k distinct elements has a group element  An action is k-fold transitive if for any k-tuples of distinct elements  there is  Definition 1.5 A group action is primitiveif there is no non-trivial partition of X preserved by the group G. A doubly transitive group action is primitive and a primitive action is transitive, but neither the, converse holds. Definition 1.6 Let G be a transitive group. A subset X of  is said to be a set of imprimitivity for the action of G on  , if for each  either  . In particular, 1- element subsets of and the empty set are obviously sets of imprimitivity of every group G on ; these are called trivial sets of imprimitivity. We say that G is primitive on  if the only sets of imprimitivity are the trivial ones; otherwise G is imprimitive on  Definition 1.7 The factor group of the normal subgroup N in a group G written as  is the set of cosets of N in G.  Definition 1.8 A composition series for a group G is a finite chain of subgroups  such that, for   is a normal subgroup of  and the quotient group   is simple. The quotient groups   are called the composition factors of G. Example   Let  , and consider the following chain  of  subgroups:  . . . . (4.1) We know that and  . Since  is an abelian group, . Certainly . Hence (4.1) is a series of subgroups, each normal in the previous one. We can calculate the order of each subgroup, and hence calculate the order of the quotient groups:     Thus the quotients are all of prime order. We now make use of the fact that a group G of prime order p is both cyclic and simple (see Example 3.6), to see that the factors for the series (4.1) are cyclic simple groups. Thus (4.1) is composition series for S4, with composition  factors   Definition 1.9 Let G be a group. A subnormal series of G is a finite chain of subgroups    such that  is a normal subgroup of  for   . The collection of quotient groups are the factors of the series, and the length of the series is n. Note that we do not require each subgroup in the subnormal series to be normal in the whole group, only that it is normal in the previous subgroup in the chain.  A normal series is a series where  is a normal subgroup of G for all i. Note also that the length n is also the number of factors occurring. We will be interested in three different types of subnormal series in this research, and for all three we will require special properties of the factors. The first case is where the factors are all required to be simple groups. Definition 1.10 The series of subgroups  Such that  where  is abelianis called a solvable series.  Definition 1.11 (Milne, J.S, 2009) Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.7 No.24 2017  80 A group G is solvable if there is a finite collection of groups  such that  where  and is abelian.If   then Gis considered as solvable group. Theorem 1.1 (Audu M.S, 2003) Let C and D be permutation groups on Гand respectively. Let be the set of all maps of  into the permutation group C. That is in _. Let in be defined in by  With respect to this operation of multiplication acquire a structure of a group.  Proof  (i)   is non-empty and is closed with respect to multiplication. If then  . Hence  . This implies that  and so  (ii) Since multiplication is associative so also is the multiplication in .  (iii) The identity element in  is the map e : ∆ → C given by e(δ)=1 for all δ א ∆ and 1 א C.  (iv) Every element f א is defined for all δ א ∆ by . Thus C∆ is a group with respect to the multiplication defined above. (We denote this group by P ). Lemma 1.2 (Audu M.S, 2003) Assume that D acts on P as follows: . Then D acts on P as a group.  Proof  Take and  then  (i)  (ii)  (iii) . Thus D acts on P as a group  Theorem 1.3 (Audu M.S, 2003) Let D act on P as a group. Then the set of all ordered pairs  with  and  acquires the structure of a group when we define for all  Proof  (i) Closure property follows from the definition of multiplication.  (ii) Take . Then        Also we have in the same manner that        .  hence multiplication is associative.  (iii) We know that for every . Now for every  the map is an automorphism of P. Also if e is the identity element in P then . Also . Now  Also using the reverse order we have that   Thus identity element exists.  (iv)  Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.7 No.24 2017  81 Also      Thus when D acts on P the set of all ordered pairs  is a group if we define In what follows we supply a formal definition of Wreath Product of permutation groups. WREATH PRODUCT (Audu M.S, 2003) The Wreath product of C by D denoted by W = C wr D is the semidirect product of P by D so that  with multiplication in W defined as  for all . Henceforth we write f d instead of  for elements of W.  Theorem 1.4 (Audu M.S, 2003) Let D act on P as  where  Let W be the group of all juxtaposed symbols f d with  and multiplication given by  Then W is a group called the semi-direct product of P by D with the defined action.  Based on the forgoing we note the following: v If C and D are finite groups then the wreath product W determined by an action of D on a finite set is a finite group of order v P is a normal subgroup of W and D is a subgroup of W. v The action of W on Γ × ∆ is given by  We shall at this point identify the conditions under which a sup group will be soluble or nilpotent and study them for further investigation. Theorem 1.5 (Thanos G., 2006) is solvable if and only if  for some n. Proposition 2.1 Let G be solvable and . Then  1. H is solvable.  2. If , then  is solvable.  Proof  Start from a series with abelian slices. . Then . When H is normal, we use the canonical projection  to get  the quotients are abelian as well, so  is still solvable. Proposition 1.6 Let . Then  is solvable if and only if  and  are solvable.  Proof (֜) This is obvious by Proposition 2.1.  (֚) Stick together a series for  with abelian slices with the lift to G of a series for , using the fourth isomorphism law.  RESULT 2.1 Symmetric groups 2.1.1 Consider the symmetric group acting on     with primitive perfect residuum   2.1.2 Consider the symmetric group acting on   with primitive perfect residuum  2.1.3 Consider the symmetric group acting on   , Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.7 No.24 2017  82     with primitive perfect residuum  2.2 Alternating group 2.2.1 Consider the alternating group acting on     with primitive perfect residuum   2.2.2 Consider the alternating group acting on   with primitive perfect residuum  2.2.3 Consider the alternating group acting on     with primitive perfect residuum  2.3 Dihedral group 2.3.1 Consider the dihedral group acting on     with primitive perfect residuum  2.3.2 Consider the dihedral group acting on     with primitive perfect residuum  2.3.3 Consider the dihedral group acting on    with primitive perfect residuum 2.4 Wreath product 2.4.1 Consider the permutation groups   and   acting on the sets  respectively. Let Let  We can easily verify that is a group with respect to the operations   . The wreath product of  and  is given by  , where  with imprimitive perfect residuum  2.4.2 Consider the permutation groups   and     acting on the sets  respectively. Let Let  We can easily verify that is a group with respect to the operations   . The wreath product of  and  is given by  , where  Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.7 No.24 2017  83                        Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.7 No.24 2017  84                            Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.7 No.24 2017  85               with imprimitive perfect residuum    SUMMARY OF RESULT Ø The perfect residuum of a solvable group is always identity while for unsolvable group is not trivial. Ø The perfect residuum of permutation group is prirmitive  3.1 VALIDATION 3.1.1  Algorithm for the result in 2.1.1 gap> S3:=SymmetricGroup(3); Sym( [ 1 .. 3 ] ) gap> P1:=PerfectResiduum(S1); Group(()) gap> IsPrimitive(P1); true gap>quit;  3.1.2  Algorithm for the result in 2.1.2 gap> S4:=SymmetricGroup(4); Sym( [ 1 .. 4 ] ) gap> P2:=PerfectResiduum(S4); Group(()) gap> IsPrimitive(P2); true gap>quit;  3.1.3  Algorithm for the result in 2.1.3 Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.7 No.24 2017  86 gap> S5:=SymmetricGroup(5); Sym( [ 1 .. 5 ] ) gap> P3:=PerfectResiduum(S5); Alt( [ 1 .. 5 ] ) gap> IsPrimitive(P3); true gap>quit;  3.1.4  Algorithm for the result in 2.2.1 gap> A3:=AlternatingGroup(3); Alt( [ 1 .. 3 ] ) gap> P4:=PerfectResiduum(A3); Group(()) gap> IsPrimitive(P4); true gap>quit;  3.1.5  Algorithm for the result in 2.2.2 gap> A4:=AlternatingGroup(4); Alt( [ 1 .. 4 ] ) gap> P5:=PerfectResiduum(A4); Group(()) gap> IsPrimitive(P5); true gap>quit;   3.1.6  Algorithm for the result in 2.2.3 gap> A5:=AlternatingGroup(5); Alt( [ 1 .. 5 ] ) gap> P6:=PerfectResiduum(A5); Alt( [ 1 .. 5 ] ) gap> IsPrimitive(P6); true gap>quit;  3.1.7  Algorithm for the result in 2.3.1 gap> D6:=DihedralGroup(IsGroup,6); Group([ (1,2,3), (2,3) ]) gap> P7:=PerfectResiduum(D3); Group(()) gap> IsPrimitive(P7); true gap>quit;  3.1.8  Algorithm for the result in 2.3.2 gap> D8:=DihedralGroup(IsGroup,8); Group([ (1,2,3,4), (2,4) ]) gap> P8:=PerfectResiduum(D2); Group(()) gap> IsPrimitive(P8); true gap>quit;  3.1.9  Algorithm for the result in 2.3.3 gap> D14:=DihedralGroup(IsGroup,14); Group([ (1,2,3,4,5,6,7), (2,7)(3,6)(4,5) ]) gap> P9:=PerfectResiduum(D4); Group(()) gap> IsPrimitive(P9); true gap>quit;   Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.7 No.24 2017  87 3.1.10  Algorithm for the result in 2.4.1 gap> M1:=Group((1,2,3)); Group([ (1,2,3) ]) gap> L1:=Group((4,5)); Group([ (4,5) ]) gap> W1:=WreathProduct(M1,L1); Group([ (1,2,3), (4,5,6), (1,4)(2,5)(3,6) ]) gap> P10:=PerfectResiduum(W1); Group(()) gap> IsPrimitive(P10); true gap>quit;  3.1.11  Algorithm for the result in 2.4.2 gap> M2:=Group((1,2,3,4,5)); Group([ (1,2,3,4,5) ]) gap> L2:=Group((6,7,8)); Group([ (6,7,8) ]) gap> W2:=WreathProduct(M2,L2); Group([ (1,2,3,4,5), (6,7,8,9,10), (11,12,13,14,15), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15) ]) gap> P11:=PerfectResiduum(W2); Group(()) gap> IsPrimitive(P11); true gap>quit;   Reference 1. Audu M.S., K.E Osondu, A.R.j. Solarin (2003),”Research Seminar on Groups, Semi Groups and Loops,” National Mathematical Centre, Abuja, Nigeria (October). 2. Burnside W. (2012),”Theory of Groups of Finite Order,” Ebooks $ 40395. 3. Hall M. Jr. (1959),” The Theory of Groups” Macmillan Company New York. 4. Hamma S. and Mohammed M.A. (2010),”Constrcting p-groups From Two Permutation Groups by Wreath Products Method”,Advances in Applied Science Research, 2010, 1(3), 8-23 5. Hielandt, H. (1964),” Finite Permutation Groups”, Academic Press, New York, London. 6. Thanos G. (2006), “ Solvable Groups-A numerical Approach” 7. Joachim Neubüser et al(2016), “Groups Algorithm and Programming”, 4.8.5, 25-Sep-2016, build of 2016-09-25 14:51:12 (GMTDT) 8. Kurosh A.G. (1956), "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) M. 9.  9. Bello et al. (2017), “a numerical search for polycyclic and locally nilpotent permutation groups”, International Journal of scientific and engineering research, Vol. 8, issue 8, 2017. 10. Milne J.S.  (2009),” Theory of groups” September 21, 2009 11. William Burnside (2012)’’Theoryof Groups of Finite order’’Ebook$40395.  

PRIMITIVITY OF PERFECT RESIDUUM OF PERMUTATION GROUPS

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PRIMITIVITY OF PERFECT RESIDUUM OF PERMUTATION GROUPS

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