Finite groups with quasi-dihedral and wreathed Sylow $2$-subgroups.
HTML articles powered by AMS MathViewer
- by J. L. Alperin, Richard Brauer and Daniel Gorenstein PDF
- Trans. Amer. Math. Soc. 151 (1970), 1-261 Request permission
Abstract:
The primary purpose of this paper is to give a complete classification of all finite simple groups with quasi-dihedral Sylow 2-subgroups. We shall prove that any such group must be isomorphic to one of the groups ${L_3}(q)$ with $q \equiv - 1 \pmod 4,{U_3}(q)$ with $q \equiv 1 \pmod 4$, or ${M_{11}}$. We shall also carry out a major portion of the corresponding classification of simple groups with Sylow 2-subgroups isomorphic to the wreath product of ${Z_{{2^n}}}$ and ${Z_2},n \geqq 2$.References
- J. L. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222โ241. MR 215913, DOI 10.1016/0021-8693(67)90005-1 H. Bender, Doubly transitive groups with no involution fixing two points (to appear). โ, Finite groups having a strongly embedded subgroup (to appear).
- Richard Brauer, On groups whose order contains a prime number to the first power. I, Amer. J. Math. 64 (1942), 401โ420. MR 6537, DOI 10.2307/2371693
- Richard Brauer, Zur Darstellungstheorie der Gruppen endlicher Ordnung, Math. Z. 63 (1956), 406โ444 (German). MR 75953, DOI 10.1007/BF01187950
- Richard Brauer, Some applications of the theory of blocks of characters of finite groups. I, J. Algebra 1 (1964), 152โ167. MR 168662, DOI 10.1016/0021-8693(64)90031-6
- Richard Brauer, On blocks and sections in finite groups. I, Amer. J. Math. 89 (1967), 1115โ1136. MR 219637, DOI 10.2307/2373422
- Richard Brauer, On finite Desarguesian planes. I, Math. Z. 90 (1965), 117โ123. MR 193152, DOI 10.1007/BF01112235
- R. Brauer and C. Nesbitt, On the modular characters of groups, Ann. of Math. (2) 42 (1941), 556โ590. MR 4042, DOI 10.2307/1968918
- Richard Brauer and Michio Suzuki, On finite groups of even order whose $2$-Sylow group is a quaternion group, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1757โ1759. MR 109846, DOI 10.1073/pnas.45.12.1757
- Richard Brauer and Hsio-Fu Tuan, On simple groups of finite order. I, Bull. Amer. Math. Soc. 51 (1945), 756โ766. MR 15102, DOI 10.1090/S0002-9904-1945-08441-9
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
- Roger Carter and Paul Fong, The Sylow $2$-subgroups of the finite classical groups, J. Algebra 1 (1964), 139โ151. MR 166271, DOI 10.1016/0021-8693(64)90030-4
- Jean Dieudonnรฉ, La gรฉomรฉtrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 5, Springer-Verlag, Berlin-Gรถttingen-Heidelberg, 1955 (French). MR 0072144
- Walter Feit and John G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775โ1029. MR 166261
- Paul Fong and W. J. Wong, A characterization of the finite simple groups $\textrm {PSp} (4,q)$, $G_{2}(q)$, $D_{4}{}^{2}(q)$. I, Nagoya Math. J. 36 (1969), 143โ184. MR 255666, DOI 10.1017/S0027763000013180
- George Glauberman, A characteristic subgroup of a $p$-stable group, Canadian J. Math. 20 (1968), 1101โ1135. MR 230807, DOI 10.4153/CJM-1968-107-2
- George Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403โ420. MR 202822, DOI 10.1016/0021-8693(66)90030-5
- Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
- Daniel Gorenstein, Finite groups the centralizers of whose involutions have normal $2$-complements, Canadian J. Math. 21 (1969), 335โ357. MR 242939, DOI 10.4153/CJM-1969-035-x
- Daniel Gorenstein and John H. Walter, On finite groups with dihedral Sylow 2-subgroups, Illinois J. Math. 6 (1962), 553โ593. MR 142619 โ, The characterization of finite groups with dihedral Sylow 2-subgroups. I, II, III, J. Algebra 2 (1965), 85-151, 218-270, 354-393. MR 31 #1297a, b; MR 32 #7634.
- J. A. Green, Blocks of modular representations, Math. Z. 79 (1962), 100โ115. MR 141717, DOI 10.1007/BF01193108
- Koichiro Harada, Finite simple groups with short chains of subgroups, J. Math. Soc. Japan 20 (1968), 655โ672. MR 230811, DOI 10.2969/jmsj/02040655 โ, A characterization of ${U_3}(5)$, Nagoya J. Math. 38 (1970), 27-40. M. OโNan, A characterization of ${U_3}(q)$, q odd, Doctoral Thesis, Princeton University, Princeton, N. J., 1969. I. Schur, รber eine Klasse von endlichen Gruppen linearer Substitutionen, S.-B. Preuss. Akad. Wiss. Berlin 1905, 77-91. โ, Untersuchungen รผber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutione, Crelle J. 132 (1907), 85-137.
- Robert Steinberg, The representations of $\textrm {GL}(3,q), \textrm {GL} (4,q), \textrm {PGL} (3,q)$, and $\textrm {PGL} (4,q)$, Canad. J. Math. 3 (1951), 225โ235. MR 41851, DOI 10.4153/cjm-1951-027-x
- Michio Suzuki, A characterization of the $3$-dimensional projective unitary group over a finite field of odd characteristic, J. Algebra 2 (1965), 1โ14. MR 178078, DOI 10.1016/0021-8693(65)90021-9
- John G. Thompson, Defect groups are Sylow intersections, Math. Z. 100 (1967), 146. MR 213432, DOI 10.1007/BF01110791
- W. J. Wong, On finite groups whose $2$-Sylow subgroups have cyclic subgroups of index $2$, J. Austral. Math. Soc. 4 (1964), 90โ112. MR 0160816, DOI 10.1017/S1446788700022771
- Hans J. Zassenhaus, The theory of groups, Chelsea Publishing Co., New York, 1958. 2nd ed. MR 0091275
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 151 (1970), 1-261
- MSC: Primary 20.27
- DOI: https://doi.org/10.1090/S0002-9947-1970-0284499-5
- MathSciNet review: 0284499