صد سال با مرکزساز عضوهای یک گروه

نوع مقاله : مقاله مروری

نویسندگان

1 دانشگاه زنجان، دانشکده علوم، گروه ریاضی

2 آموزش و پرورش استان زنجان، پژوهشگاه تعلیم و تربیت ملاصدرا، دبیرخانه کشوری ریاضی

چکیده

مطالعۀ تاریخ ریاضی علاوه بر ادای دین نسبت به پیشگامان و تلاشگران این حوزۀ علمی، نحوۀ تکامل موضوعات ریاضی را نیز آشکار می کند. به ویژه ریاضی خوانان جوان طی مطالعۀ تاریخ ریاضی با جریان فکری حاکم بر فرآیند کشف قضیه های ریاضی آشنا می شوند و خود می توانند مستقلا به بازآفرینی روند حل مسائل و یادگیری عمیق ریاضی بپردازند. در این مقاله، تاریخچه ای از تعریف و به کارگیریِ مرکزساز عضوهای گروه ها را در شناسایی ساختار انواع گروه ها ارائه می دهیم. به ویژه به مهم ترین نقش آفرینی های مرکزساز عضوها در نظریه گروه ها نظیر طبقه بندی گروه های سادۀ متناهی اشاره می کنیم

کلیدواژه‌ها

موضوعات


قریشی، سید محسن، نظریۀ گروه ها: سرگذشت و سرنوشت، ریاضی و جامعه، سال ٣ ١٣٩٧ ،شماره ١ ،١٧-٩.
لم، ت. ی.، صد سال نمایش نظریۀ گروه ها (بخش اول)، ترجمه علیرضا جمالی، نشر ریاضی، سال ١٠ ١٣٧٨ ،شماره ١٩ ٢١-٣٢ .
لم، ت. ی.، صد سال نمایش نظریۀ گروه ها (بخش دوم)، ترجمه علیرضا جمالی، نشر ریاضی، سال ١٠) ١٣٧٨ ،(شماره ٢٠، ۵۵-۶۴ . 
Abdollahi, A., Jafarian Amiri S. M., Mohammadi Hassanabadi, A., Groups with specific number of centralizers, Houston J. Math., 33 (2007), no. 1, 43–57.
Abdollahi A., Mohammadi Hassanabadi, A., Non-cyclic graph associated with a group, J. Algebra Appl., 8 (2009), no. 2, 243–257.
Abdollahi A., Mohammadi Hassanabadi, A., Noncyclic graph of a group, Comm. Algebra, 35 (2007), no. 7, 2057–2081.
Abdollahi, A., Zarrin, M., Non-nilpoten graph of a group, Comm. Algebra, 38 (2010), no. 12, 4390–4403.
Arad, Z., A classification of groups with a centralizer condition, Bull. Aust. Math. Soc., 15 (1976), no. 1, 81–-85.
Arad Z., Chillag, D., On finite groups containing a CC-subgroup, Arch. Math. (Basel), 29 (1977), no. 3, 225–234.
Arad Z., Herfort, W., Classification of finite groups with a CC-subgroup, Comm. Algebra, 32 (2004), no. 6, 2087–2098.
Arad Z., Chillag, D., Finite groups containing a nilpotent Hall subgroup of even order, Houston J. Math., 7 (1981), no. 1, 23–-32.
Arad Z., Herfort, W., A classification of locally finite and profinite groups with a centralizer condition, Comm. Algebra, 10 (1982), no. 16, 1749–1764.
Arad Z., Herfort, W., The history of the classification of groups containing a CC-subgroup, Contemp. Math., 402 (2006), 1–12.
Arad, Z., Herzog, M., A classification of groups with a centralizer condition II: Corrigendum and addendum, Bull. Aust. Math. Soc., 17 (1 ) (1977), 157–160.
 Aschbacher, M., Bender, H., Feitو W., Solomon, R., Michio Suzuki (1926–1998), Notices Amer. Math. Soc., 46 (1999), no. 5, 543–551.
Aschbacher, M., Daniel Gorenstein (1923-1992), Biographical Memoir of National Academy of Sciences, (2016), 1–17.
Ashrafi, A. R., On finite groups with a given number of centralizers, Algebra Colloq.,7 (2) (2000), 139–146.
Ashrafi, A. R., Counting the centralizers of some finite groups, Korean J. Comput. Appl. Math., 7 (1) (2000), 115–124.
Ashrafi A. R., Taeri, B., On finite groups with a certain number of centralizers, J. Appl. Math. Comp., 7 (2005), 217–227.
Baishya, S. J., On finite groups with specific number of centralizers, Int. Elec. J. Algebra, 13 (2013), 53–62.
Belcastro, S. M., Sherman, G. J., Counting centralizers in finite groups, Math. Mag., 5 (1994), 111–114.
Bertram, E. A., Large centralizers in finite solvable groups, Israel J. Math., 47 (4) (1984), 335–344.
Bertram, E. A., Herzog, M., Finite groups with large centralizers, Bull. Aust. Math. Soc., 32 (3) (1985), 399-414.
Brauer, R., Fowler, K., On groups of even order, Ann. Math. 62 (1955), 565–583.
Burns, J. E., The foundation period in the history of group theory, Amer. Math. Monthly, 20 (5) (1913), 141–148.
Busarkin, V. M., Structure of strongly isolated subgroups of finite groups, Algebra Logika,4 (2) (1965), 33–50 .
Casolo, C., Finite groups in which subnormalizers are subgroups, Rend. Semin. Mat. Univ. Padova, 82 (1989), 25–53.
Casolo, C., Subnormalizers in finite groups, Comm. Algebra, 18 (11) (1990), 3791–3818.
Cossey, J., Finite soluble groups have large centralisers, Bull. Aust. Math. Soc., 35 (1987), 291–298.
Dolfi, S., Herzog M., Jabara, E., Finite groups whose noncentral commuting elements have centralizers of equal size, Bull. Aust. Math. Soc., 82 (2010), 293–304.
Dolfi, S., Jabara, E., Lucido, S., C55-Groups, Sib. Math. J., 45 (6) (2004), 1053–1062.
Dutta, J., Basnet, D. K., Nath, R. K., A note on n-centralizer finite rings, arXiv:1512.00973, 2015.
Feit, W., On the structure of Frobenius groups, Canad. J. Math., 9 (1957), 587–596.
Feit, W., On groups which contain Frobenius groups as subgroups, Proc. Sympos. Pure Math., 1, Amer. Math. Soc., Providence, RI, 1959, 22–28.
Feit, W., On a class of doubly transitive permutation groups, Illinois J. Math., 4(1960), 170–186.
Feit, W., A characterization of the simple groups SL(2, 2), Amer. J. Math., 82 (1960), 281–300.
Feit, W., Correction: A characterization of the simple groups SL(2, 2)a), Amer. J. Math., 84
(1962), 201–204.
Foruzanfar, Z., Mostaghim, Z., On 10-Centralizer Groups of Odd Order, ISRN Algebra, Volume 2014, Article ID.: 607984 (4 pages).
Freese, R., A Review of Subgroup Lattices of Groups, by Roland Schmidt, Wed. Feb. 28 13:55:46 HST 1996, http://www.math.hawaii.edu/ ralph/schmidt/sch-protter/sch-protter.html.
Gallian, J. A., Classification of finite simple groups completed, MAA Focus, 1 (1981), 3–7.
Green, J. A., Richard Dagobert Brauer, Bull. London Math. Soc., 10 (1978), no. 3, 317–342.
Harada, K., Michio Suzuki, in Groups and Combinatorics in memory of Michio Suzuki, special issue, Advanced Studies in Pure Mathematics, 32 (2001), 1–39.
Heineken, H., On E-groups in the sense of Peng, Glasg. Math. J., 31 (1989), 231–242.
Herzog, M., On finite groups which contain a Frobenius group, J. Algebra, 6 (1967), 192–221.
Herzog, M., On finite simple groups of order divisible by three primes only, J. Algebra, 10 (1968), 383–388.
Hoseiniravesh, M., Rajabzadeh Moghaddam, M. R., Derakhshandeh, M. F., Lie algebras with few centralizers, Comm. Algebra, 45 (2017), no. 7, 2867-2874.
http://www-history.mcs.st-andrews.ac.uk/Biographies/Cipolla.html, accessed April 17, 2017.
Iiyori, N., Yamaki, H., Prime Graph Components of the Simple Groups of Lie Type over the Field of Even Characteristic, J. Algebra, 155 (1993), no. 2, 335–343.
Iiyori, N., Yamaki, H., Corrigendum to: prime graph components of the simple groups of Lie type over the field of even characteristic, J. Algebra, 181 (1996), 659–660.
Isaacs, I. M., Solvable groups contain large centralizers, Israel J. Math., 55 (1986), no. 1, 58–64.
Ito, N., On finite groups with given conjugate type, I, Nagoya J. Math., 6 (1953), 17–28.
Jafarian Amiri, S. M., Rostami, H., Finite groups all of whose proper centralizers are cyclic, Bull. Iranian Math. Soc., 43 (2017), no. 3, 755–762.
 Jafarian Amiri, S. M., Amiri, M., Madadi M., Rostami, H., Finite groups have even more centralizers, Bull. Iranian Math. Soc., 41 (2015), no. 6, 1423–1431.
Jafarian Amiri, S. M., Amiri, M., Madadi, M., Rostami, H., On 9-centraliser groups, J. Algebra Appl., 14 (2015), no. 1, 1550003 (13 pages).
Jafarian Amiri, S. M., Amiri, M., Madadi, M., Rostami, H., On the probability of generating nilpotent subgroups in a finite group, Bull. Aust. Math. Soc., 93 (2016), 447–453.
Jafarian Amiri, S. M., Amiri, M., Madadi, M., Rostami, H., On F-groups with central factor of order p4 , Math. Slovaca, 67 (5) (2017), 1147–1154.
Jafarian Amiri, S. M., Rostami, H., Groups with a few nonabelian centralizers, Publ. Math. Debrecen, 87 (2015), no. 3-4, 429–437.
Jafarian Amiri, S. M., Amir M., Rostami, H., Finite groups determined by the number of element centralizers, Comm. Algebra, 45 (2017), no. 9, 3792–3797.
Jafarian Amiri, S. M., Rostami, H., Centralizers and the maximum size of the pairwise noncommuting elements in finite groups, Hacett. J. Math. Stat., 46 (2017), no. 2, 193–198.
Jafarian Amiri, S. M. Rostami, H., Finite groups in which every centralizer of the noncentral element of odd order is abelian, J. Algebra Appl., 18 (2019), no. 6, 1950108 (7 pages).
Jafarian Amiri, S. M., Madadi, M., Rostami, H., Groups with exactly ten centralizers, Bull. Iranian Math. Soc., 44 (2018), 1163–1170.
Jafarian Amiri, S. M. Rostami, H., Centralizers in a group whose central factor is simple, J. Algebra Appl., 17 (2018), no. 8, 1850149.
Kleiner, I., The evolution of group theory: A brief survey, Math. Mag., 59 (1986), no. 4, 195–215.
Kondrat’iev, A. S., Prime graph components of finite simple groups, Math. USSR Sb., 67 (1990), no. 1, 235–247. Translation from Mat. Sb., 180 (1989), no. 6, 787–797.
Kondrat’iev„ A. S., Mazurov, V. D., Recognition of alternating groups of prime degree from the orders of their elements, Sibirsk Mat. Zh., 41, no. 2, 359–369 (Russian). Translation in Sib. Math. J., 41 (2000), no. 2, 294–302.
Kosvintsev, L. F., Finite groups with maximal element centraliers, Math. Notices Acad. Sci. USSR, 13 (1973), no. 4, 577-580.
Lucido, M. S., Prime graph components of finite almost simple groups, Rend. Semin. Mat. Univ. Padova, 102 (1999), 1–22.
Lucido, M. S., Addendum to prime graph components of finite almost simple groups, Rend. Semin. Mat. Univ. Padova, 107 (2002), no. 1–2, 189–190.
Maier, V. R., Finite groups in which elements of odd order have abelian centralizers, Sib. Math. J., 16 (1963), no. 3, 423–430.
Miller, G. A., Group theory in the history of mathematics, Sci. Monthly, 47 (1938), no. 2, 124–127.
Mousavi, L., n-Cyclicizer groups, Bull. Iranian Math. Soc., 37 (2011), 161–170.
 Nasrabadi, M. M., Gholamian, A., On Finite n-Acentralizer Groups, Comm. Algebra, 43 (2015), no. 2, 378–383.
Peng,T. A., On groups with nilpotent derived groups, Arch. Math., 20 (1969), 251–253.
Peng,T. A., Finite soluble groups with an Engel condition, J. Algebra, 11 (1969), 319–330.
Rajabzade Moghaddam, M. R., Rostamyari, M. A., 2-Engelizer subgroup of a 2-Engel transitive groups, Bull. Korean Math. Soc., 53 (2016), no. 3, 657–665.
Rebmann, J., F-Grouppen, Arch. Math., 22 (1971), 225–230.
Redei, L., Ein Satz über die endlichen einfachen Gruppen, Acta. Math., 84 (1950), 129–153.
Saeedi, F., Farrokhi, M., Finite groups with a given number of relative centralizers, Comm. Algebra, 46 (2018), no. 1, 378–385.
Schmidt, R., Zentralisatorverbande endlicher Gruppen, Rend. Sem. Mat. Univ. Padova, 44 (1970), 97–131.
Scott, L., Solomon, R., Thompson, J., Walter, J., Zelmanov, E., Walter Feit (1930-2004), Notices Amer. Math. Soc., 52 (2005), no. 7, 728–735.
Suzuki, M., Structure of a Group and the Structure of its Lattice of Subgroups, Springer-Verlag, Berlin, 1956.
Suzuki, M., On a class of doubly transitive groups, Ann. of Math., 75 (1962), 105–145.
Suzuki, M., The nonexistence of a certain type of simple groups of odd order, Proc. Amer. Math. Soc., 8 (1957), 686-695.
Suzuki, M., On characterizations of linear groups, I, II, Trans. Amer. Math. Soc., 92 (1959), 191–219.
Thompson, J. G., Normal p-complements for finite groups, J. Algebra, 1 (1964), 43–46.
Tolue, B., The non-centralizer graph of a finite group, Math. Rep., 17 (2015), no. 3, 265–275.
Vasil’eva, A. V., Centralizer lattices of finite simple groups, Sib. Math. J., 18 (1977), no. 2, 251–270 .
Vasil’eva, A. V., Characterization of the group P SL(2, q) by its centralizer lattice, Algebra Logika, 15 (1976), no. 5, 509–53.
Weisner, L., Group-theoretic origin of certain generating functions, Pacific J. Math., 5 (1955), 1033–1039.
Weisner, L., Groups in which the normalizer of every element except the identity is abelian, Bull. Amer. Math. Soc., 31 (1925), 413–416.
Williams, J. S., Prime graph components of finite groups, J. Algebra, 69 (1981), 487–513.
Zarrin, M., Criteria for the solubility of finite groups by its centralizers, Arch. Math., 96 (2011), 225–226.
Zarrin, M., On element centralizers in finite groups, Arch. Math., 93 (2009), 497–503.
Zarrin, M., On solubility of groups with finitely many centralizers, Bull. Iranian Math. Soc., 39 (2013), 517–521.
Zarrin, M., On non-commuting sets and centralisers in infinite groups, Bull. Aust. Math. Soc., 93 (2016), no. 1, 42–46.
Zarrin, M., On noncommuting sets and centralizers in finite groups, Bull. Aust. Math. Soc., 10 (2015), 1–5.
Zarrin, M., Derived length and centralizers of groups, J. Algebra Appl., 14 (2015), no. 8, 1550133.