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Publications

Our current research interest is concentrated in three fields, namely, Group Theory, LA-Semigroup Theory and History of Mathematics. The papers are listed in chronological order. Mathematical Review numbers are given at the end of each paper reviewed in the Mathematical Reviews published by the American Mathematical Society, USA.

Q.Mushtaq and S.M. Yusuf, On LA-semigroups, The Aligarh Bull. Math., 8(1978), 65-70.
MR 84c:20086
Q.Mushtaq and S.M. Yusuf, On Locally Associative LA-semigroups, J.Nat.Sci. Math., 19,1(1979), 57-62.
MR 82a:20081
Q.Mushtaq, Abelian Groups Defined by LA-semigroups, Studia Sci. Math. Hungar., 18(1983), 427-428.
MR 86h:20106
G. Higman and Q.Mushtaq, Coset Diagrams and Relations for PSL(2,Z), Arab Gulf J. Scient. Res., 1, 1(1983), 159-164. (Saudi Arabia)
MR 85d:11044
Q.Mushtaq, Al-Khowarizmi’s Legacy in Algebra, J. Central Asia, 10,1 (1987), 31-36.
Q.Mushtaq and F.Shaheen, Coset Diagrams for a Homomorphic Image of D(2,3,6), Ars Combinatoria, 23A(1987), 187-193.
MR 8hc:20038
Q.Mushtaq, Some Remarks on Coset Diagrams for the Modular Group, Math. Chronicle, 16 (1987), 69-77.
MR 89f:11066
Q.Mushtaq and S.M. Yusuf, On LA-semigroup Defined by a Commutative Inverse Semigroup, Mat. Vesnik., 40(1988), 59-62.
MR 89k:20102
Q.Mushtaq, Modular Group Acting on Real Quadratic Fields, Bull. Austral. Math. Soc., 37 (1988), 303-309.
MR 88e:11065
Q.Mushtaq, A Condition for the Existence of a Fragment of a Coset Diagram, Quart, J. Math. Oxford, 39, 2(1988), 81-95.
MR 89b:20078
Q.Mushtaq, Science in the Mediterranean Before Advent of Islam, J. Central Asia, 11,2 (1988), 161-167.
Q.Mushtaq, A Note on Almost Semigroups, Bull. Malaysian Maths. Soc. (2nd Ser.), 11(1988), 29-31.
MR 90f:20110
Q.Mushtaq, Reduced Indefinite Binary Quadratic Forms and Orbits of the Modular Group, Radovi Matematicki, 4(1988), 331-336.
MR 90d:11044
Q.Mushtaq, A Note on Translative Mappings on LA-semigroups, Bull. Malaysian Math. Soc. (2nd Ser.), 11 (1988), 39-42.
MR 90j:20147
Q.Mushtaq, Trigonometry as an Independent Science, J. Central Asia, 12,1 (1989), 39-45.
Q.Mushtaq and M.S.Kamran, On LA-semigroups with Weak Associative Law, Sci. Khyber, 2,1 (1989), 69-71.
Q.Mushtaq, Coset Diagrams for an Action of the Extended Modular Group on the Projective Line over a Finite Field, Indian J. Pure and Applied Math., 20, 8(1989), 747-754.
MR 90j:20113
Q.Mushtaq, The Extended Modular Group Acting on the Projective Line over a Galois Field, Indian J. Pure and Applied Math., 20, 8(1989), 755-760.
MR 90j:20114
Q.Mushtaq, Embedding of an LA-semigroup in a Commutative Monoid, Proc. Pak. Acad. Sci., 26,4 (1989), 319-324.
MR 90m:20073
Q.Mushtaq, Muslim Legacy in Number Theory, J. Central Asia, 12,2 (1989), 73-82.
Q.Mushtaq, Medieval Muslim Geometry, J. Central Asia,12,2 (1989), 99-108.
Q.Mushtaq, Influence and Transfer of Islamic Science to Medieval Europe, Islamic Thought and Creativity, 1, 2 (1990), 5-16.
Q.Mushtaq and F.Shaheen, On the Subgroups of Polyhedral Groups, Math. Japonica, 35, 5(1990), 849-853.
MR 91j:20123
Q.Mushtaq and Q. Iqbal, Decomposition of a Locally Associative LA-semigroup, Semigroup Forum, 41(1990), 155-164.
MR 91f:20067
Q.Mushtaq, Coset Diagrams for Hurwitz Groups, Communications in Algebra, 18, 11(1990), 3857-3888.
MR 91j:20122
Q.Mushtaq and M.Iqbal, Partial Ordering and Congruence on LA-semigroups, Ind. J. Pure & App. Math., 22,4(1991), 331-336.
MR 92f:20085
Q.Mushtaq and A.L.Tan, Evolution of Science During the 8th to 11th Century A.D., Islamic Thought and Creativity, 2, 2(1991), 7-22.
Q.Mushtaq and Q.Iqbal, On Semilattice Structure of LA-semigroups, Proc. Pak. Acad. Sci., 28, 2(1991), 197-200.
MR 93i:20083
Q.Mushtaq, M.Ashiq and T.Maqsood, The Symmetric Groups as Quotients of G3,8,720, Math. Japonica, 37, 1(1992), 9-16.
MR 83a:20046
Q.Mushtaq and F. Shaheen, Some Special Circuits in Coset Diagrams, Math. Japonica, 37, 1(1992), 149-158.
MR 92m:20044
Q.Mushtaq, Parametrization of all Homomorphisms from PGL(2,Z) into PGL(2,q), Communications in Algebra, 20, 4(1992), 1023-1040.
MR 92m:20041
Q.Mushtaq, Chinese Mathematics and its Influence on Muslim World, Islamic Thought and Creativity, 3, 4(1992), 19-26.
Q.Mushtaq and Gian-Carlo Rota, Alternating Groups as Quotients of two Generator Groups, Advances in Math., 96, 1(1992), 113-121.
MR 93j:20067
Q.Mushtaq, M. Ashiq and T. Maqsood, Factor Groups of G5,5,120, Communications in Algebra, 20, 12(1992), 3759-3767.
MR 94c:20009
Q.Mushtaq and M.Iqbal, Ternary Operations and LA-semigroups, Proc. Pak. Acad. Sci., 29, 4(1992), 311-317.
MR 93j:20129
Q.Mushtaq and M.Aslam, Group Generated by Two Elements of Orders Two and Four Acting on Real Quadratic Fields, Acta. Math. Sinica, 9, 1(1993), 48-54.
MR 94g:20067
Q.Mushtaq and F. Shaheen, On Graphs for Certain Polyhedral Groups, Journal of Algebra, 159(1993), 332-342.
MR 94i:20068
Q.Mushtaq and H. Servatius, Permutation Representations of the Symmetric Groups of Regular Hyperbolic Tessellations, Jour. London Math. Soc., 48, 2(1993), 77-86.
MR 94h:20002
Q.Mushtaq and M. Iqbal, On Representation Theorem for Inverse LA-semigroups, Proc. Pak. Acad. Sci., 30, 4(1993), 247-253.
Q.Mushtaq and F. Shaheen, Factor Groups of G6,6,6 through Coset Diagrams for an Action on PL(Fq), Discrete Mathematics, 126 (1994), 225-238.
MR 94m:20069
Q.Mushtaq, M. Ashiq and T. Maqsood, The Quotients of G5,7,84, Math. Japonicae, 40, 2 (1994), 349-353.
MR 95i:20052
Q.Mushtaq and F. Shaheen, Finite Presentation of Alternating Groups, Acta Mathematica Sinica, 11, 2 (1995), 221-224.
MR 97e:20047
Q.Mushtaq and N.A.Zafar, Alternating and Symmetric Groups as Quotients of G5,6,36,  Proc. Intern. Conf.  On Group Theory, Pusan, Korea, Eds. A.C. Kim and D.L. Johnson, de Gruyter, Berlin, 1995, 241-247.
MR 98g:20050
Q.Mushtaq, T. Maqsood and M. Ashiq, On the Group G4,5,276, Kobe Jour. Math., 12 (1995), 1-7.
MR 96i:20040
Q.Mushtaq, Decline of Scientific Activity in the Muslim World: Causes and Remedies, Islamic Thought and Creativity, 6, 2(1995), 69-79.
Q.Mushtaq and M.S. Kamran, Translations on LA-semigroups, Proc. Pak. Acad. Sci., 31, 4 (1994), 243-249.
Q.Mushtaq and M. Ashiq, Alternating or Symmetric Groups as Quotients of G5,5,60, Algebra Colloquium, 4,3 (1996), 379-382.
MR 97i:20038
Q.Mushtaq and M.S. Kamran, On Left Almost Groups, Proc. Pak. Acad. Sci., 33, 1-2 (1996), 53-55.
MR 1423796
Q.Mushtaq and M.Aslam, Transitive Action of a Two Generator Group on Rational Projective Line, Southeast Asian Bull. Math., 1(1997), 203-207.
MR 2000a:20073
Q.Mushtaq, On Word Structure of the Modular Group over Finite and Real Quadratic Fields, Discrete Mathematics, Discrete Mathematics, 178(1998), 155-164.
MR 99b:20080
Q.Mushtaq and M.Aslam, Group Generated by two Elements of Order Two and Six Acting on Real Line and Real Quadratic Fields, Discrete Mathematics, 179(1998), 145-154.
MR 99a:20049
Q.Mushtaq and A.Ali, On Groups G5,5,180, Southeast Asian Bull. Math., 22(1998), 179-186.
MR 2000d:20043
M.S. Kamran and Q.Mushtaq, LA-groups as Quotients of Left Inverse Quasi-groups, Proc. Asian Mathematics Conference, Ratchasima-Nakhon, Thailand, 1-7, 1995. World Sci. Publ., River Edge, NJ, 1998.
MR 1660 559
Q.Mushtaq and J.L.Berggren, Introduction: the mathematicians and their heritage. History of civilizations of Central Asia, Vol. IV, 177-182, Mult. Hist. Ser., UNESCO, Paris, 2000.
MR 2002c:01015
M. Ashiq, Q.Mushtaq, T.Maqsood and M.Aslam, The Quotients of , European Journal of Combinatorics, 22,7(2001), 905-909.
MR 2002g:20057
Q.Mushtaq and M.S. Kamran, Finite AG-groupoid with Left Identity and Left Zero, Inter. Jour. of Math. & Math. Sci., 27, 6(2001), 387-389.
MR 1 870 040
Q.Mushtaq, M. Ashiq, T. Maqsood and M. Aslam, On the Coxeter Group , Communications in Algebra, 30, 5(2002), 2175-2182.
MR 2003f:20063
Q.Mushtaq, Parametrization of Triangle Groups as Subgroups of , Southeast Asian Bull. Math., 25 (2001), 309-312.
MR 1 935 102
Q.Mushtaq, Zeroids and Idempoids in AG-groupoids, Quasigroups and Related Systems, 11(2004), 79-84.
MR 2064160
Q.Mushtaq and K.Mahmood, Characterization of Division m-LA-semigroups, Quasigroups and Related Systems, 11(2004), 85-90.
MR 2064161
Q.Mushtaq and A.Mahmood, A Note on Salem numbers and Golden mean, Quasigroups and Related Systems, 11(2004), 91-93.
MR 2005b:11169
M. Ashiq and Q.Mushtaq, Parametrization of  G*(2,Z) on PL(Fq), Advances in Algebra – Proc.ICM Sattelite Conf on Algebra and Related Topics, 264 – 270, 2003.
MR2005g:20077
Q.Mushtaq, T. Maqsood, M.Aslam and M. Ashiq, Homomorphic Images of  as Subgroups of , Algebras, Groups and Geometries, 20, 4(2003), 485-499.
MR 2004m:20060
M.Aslam, Q.Mushtaq, T. Maqsood and M. Ashiq, Real Quadratic Irrational Numbers and the Group , Southeast Asian Bull. Math., 27(2003), 405-415.
MR 2005b:20097
M.Aslam and Q.Mushtaq, Closed Paths in the Coset Diagrams for  Acting on Real Quadratic Fields, Ars Combinatoria, 71(2004), 267-288.
MR 2004k:20100
N.H.Bong and Q.Mushtaq, Fibonacci and Lucas Numbers through the Action of the Modular Group on Real Quadratic Fields, Fibonacci Quart., 42, 1(2004), 20 -  27.
MR 2005h:11033
Q.Mushtaq, On Group Theory, Scientia Bruneiana, 5(2004)), 15-18,  (ISSN 1819- 9550).
Q.Mushtaq and A.Bano, Embedding of an AG**-groupoid in a commutative monoid,Quasigroups and Related Systems, 14(2004), 75-78.
MR 2005j:20074
Q.Mushtaq and M.S.Saeed, Permutation Representations of Triangle Group , Quasigroups and Related Systems, 12(2004), 75-83.
MR 2006c:20103
T.Maqsood and Q.Mushtaq, The Quotient of , Ars Combinatoria, 74(2005), 261-267.
MR:2005m:20077
Q.Mushtaq and M.S.Saeed, Action of the Group on Projective Line over Finite Fields, Southeast Asian Bulletin of Mathematics,  29(2005), 1-7.
MR 2006k:20058
M.Aslam and Q.Mushtaq, Detection of Ambiguous Numbers in the Action of the Group  on Q, Far East Jour Math., 369-378, 3(16), 2005.
MR 2006c:20105
Q.Mushtaq and T.Maqsood, Homomorphic Images of , Discrete Mathematics, 290(2005), 47-59.
MR 2006a:20064
M. Ashiq and Q.Mushtaq, Finite Presentaion of Linear Fractional Groups, Algebra Colloquium, 12, 4(2005), 585-589.
MR 2006f:20033
T. Maqsood and Q.Mushtaq, Alternating and Symmetric Groups as Homomorphic Images of , Utilitas Mathematica, 70(2006), 175-185.
MR 2238439
Q.Mushtaq and Madad Khan, Ideals in AG-band and AG*-groupoid, Quaisgroups and Related Systems, 14(2006), 207-215.
MR 2007k:20125
Q.Mushtaq and Madad Khan, Decomposition of a Locally Associative AG**-Groupoid, Advances in Algebra and Analysis, 1, 2(2006), 115-122.
MR 2007m:20109
M.Ashiq and Q.Mushtaq, Actions of a Subgroup of the Modular Group on an Imaginary Quadratic Field, Quaisgroups and Related Systems, 14(2006), 133-146.
MR 2008a:11042
Q.Mushtaq and M.Ahmed, A Graphical Technique to obtain Homomorphic Images of , Quasigroups and Related Systems, 14(2006), 195-206.
MR 2008a:20048
M.Aslam and Q.Mushtaq, Action of a linear fractional group on , Advances in Algebra and Analysis, 1, 3(2006), 159-163.
MR 2007j:11146
Q.Mushtaq and U.Hayat, Pell, Pell-Lucas numbers and the modular group, Algebra Colloquium, 14, 1(2007), 97-102.
MR:2278113
Q.Mushtaq and Madad Khan, Decomposition of AG*-groupoids, Quaisgroups and Related Systems, 15(2007), 303 – 308.
MR 2007m:20109
Q.Mushtaq and Madad Khan, A note on Abel-Grassman’s 3-band, Quaisgroups and Related Systems, 15(2007), 295 – 301.
MR 2383957
M.Ashiq and Q.Mushtaq, Actions of a 2-generator group on : The parametrization problem, Algebras, Groups and Geometries, 1, 24(2007), 187 – 202.
MR2345850
Q.Mushtaq and U.Hayat, Hardaman generalized Fibonacci numbers and the modular group, Indian J. Pure and Applied Math., 38, 5(2007), 345 – 352.
MR2361590
M.Aslam and Q.Mushtaq, Action of the group on , Southeast Asian Bulletin of Mathematics, 31, 6(2007), 1049 – 1056.
MR 2386983
M.Ashiq and Q.Mushtaq, Coset diagrams for a homomorphic image of , Acta MathScient, 28 B, 2(2008), 365 – 372.
MR 2411845
Q.Mushtaq and M.Inam, Left almost semigroups defined by a free algebra, Quaisgroups and Related Systems, 16(2008), 69 – 76.
MR2435528
Q.Mushtaq and M.Khan, Direct Product of Abel Grassmann’s Groupoids, Journal of Interdisciplinary Mathematics, 11, 4(2008), 461 – 467.
MR2444655
Q.Mushtaq and Saima Riaz, Graphical Presentation of Hurwitz Groups, Southeast Asian Bulletin of Mathematics, 32, 6(2008), 1115-1123.
Q.Mushtaq and N.Siddiqui, The Action of  on , Quaisgroups and Related Systems, 16(2008), 221 – 228.
MR2494879
Saima Anis and Q.Mushtaq, The Number of Subgroups of  when Acting on , Communications in Algebra, 36, 11(2008), 4276 – 4283.
MR2460416
Q.Mushtaq and Madad Khan, Semilattice decomposition of a locally associative AG**-Groupoid, Algebra Colloquium, 16, 1(2009), 17 – 22.
MR2477106
Q.Mushtaq and Madad Khan, M-systems in LA – semigroups, Southeast Asian Bulletin of Mathematics, 33, 6(2009), 321 – 327.
Q.Mushtaq and Sadaf Baig, A graphical technique to find the permutation representation of the triangle groups , Asian-European Jour. Math., (accepted).
T.Maqsood and Q.Mushtaq, Fragments of coset diagrams for the triangle group, Proc. 4th Int. Pure Math. Conf. 2003, (ISSN 1819-6454), Islamabad, 76 – 82.
T.Maqsood, Q.Mushtaq, M.Ashiq and M.Aslam, Homomorphic Images of  as Subgroups of , Proc. 2nd Int. Pure Math. Conf. 2001 (ISSN 1819-6454), Islamabad, 83 – 99.
Q.Mushtaq and S.Baig, Determination of triangle groups through graphs, Proc. 2nd Int. Pure Math. Conf. 2001 (ISSN 1819-6454), Islamabad, 100 – 117.
M.Ashiq and Q.Mushtaq, Transitive action of a two generator group on rational projective line, Proc. 4th Int. Pure Math. Conf. 2003, (ISSN 1819-6454), Islamabad, 1 – 8.
M.Aslam and Q.Mushtaq, Action of two generator group on an imaginary quadratic fields, Proc. 4th Int. Pure Math. Conf. 2003, (ISSN 1819-6454), Islamabad, 8 – 23.
Q.Mushtaq and S.Mustafa, Word structure of the group  over finite and real quadratic fields, 4th Int. Pure Math. Conf. 2003, (ISSN 1819-6454), Islamabad, 83 – 88.
Q.Mushtaq and S.Iqbal, Action of  and on a cubic cyclic field, Proc. 4th Int. Pure Math. Conf. 2003, (ISSN 1819-6454), Islamabad, 29 – 35.
Q.Mushtaq and Iftikhar Ali, Intransitivity of the action of   on , Proc. 6th Int. Pure Math. Conf. 2005, (ISSN 1819-6454), Islamabad, 102 – 110.
Q.Mushtaq and M.Khan, Ideals in an LA-semigroup, Proc. 4th Int. Pure Math. Conf. 2003, (ISSN 1819-6454), Islamabad, 66 – 75.
Q.Mushtaq and Shahla Asif, A5 as a homomorphic image of a subgroup of Picard group, Communications in Algebra.  38, 10 (2010), 3897-3812.
Q.Mushtaq and M.Ashiq, Parametrization of the action of a subgroup of the modular group, Canadian Jour. of Pure & Appl. Sciences
Q.Mushtaq and Nasir Siddiqui, Parametrization of actions of the automorphism subgroup of a cubic tree on PL(), Communications in Algebra.

Q.Mushtaq and Shahla Asif, Finite permutation representation of a subgroup of Picard group, Acta Math. Scient.