Characterizations of the Fischer groups. I, II, III

Parrott, David

doi:10.1090/S0002-9947-1981-0610952-5

Characterizations of the Fischer groups. I, II, III
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Trans. Amer. Math. Soc. 265 (1981), 303-347 Request permission

Abstract:

B. Fischer, in his work on finite groups which contain a conjugacy class of $3$-transpositions, discovered three new sporadic finite simple groups, usually denoted $M(22)$, $M(23)$ and $M(24)’$. In Part I two of these groups, $M(22)$ and $M(23)$, are characterized by the structure of the centralizer of a central involution. In addition, the simple groups ${U_6}(2)$ (often denoted by $M(21))$ and $P\Omega (7,3)$, both of which are closely connected with Fischer’s groups, are characterized by the same method. The largest of the three Fischer groups $M(24)$ is not simple but contains a simple subgroup $M(24)’$ of index two. In Part II we give a similar characterization by the centralizer of a central involution of $M(24)$ and also a partial characterization of the simple group $M(24)’$. The purpose of Part III is to complete the characterization of $M(24)’$ by showing that our abstract group $G$ is isomorphic to $M(24)’$. We first prove that $G$ contains a subgroup $X \cong M(23)$ and then we construct a graph (on the cosets of $X$) which is shown to be isomorphic to the graph for $M(24)$.

References

Bernd Fischer, Finite groups generated by $3$-transpositions. I, Invent. Math. 13 (1971), 232–246. MR 294487, DOI 10.1007/BF01404633
George Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403–420. 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 and Koichiro Harada, A characterization of Janko’s two new simple groups, J. Fac. Sci. Univ. Tokyo Sect. I 16 (1969), 331–406 (1970). MR 283075
Daniel Gorenstein and Koichiro Harada, On finite groups with Sylow $2$-subgroups of type $\hat A$ $_{n},\,n=8,\,9,\,10,$ and $11$, J. Algebra 19 (1971), 185–227. MR 285599, DOI 10.1016/0021-8693(71)90104-9

Finite groups whose

-subgroups are generated by at most

elements

Daniel Gorenstein and John H. Walter, On finite groups with dihedral Sylow 2-subgroups, Illinois J. Math. 6 (1962), 553–593. MR 142619
Robert L. Griess Jr., Schur multipliers of finite simple groups of Lie type, Trans. Amer. Math. Soc. 183 (1973), 355–421. MR 338148, DOI 10.1090/S0002-9947-1973-0338148-0
Robert L. Griess Jr., Schur multipliers of some sporadic simple groups, J. Algebra 32 (1974), no. 3, 445–466. MR 382426, DOI 10.1016/0021-8693(74)90151-3
Koichiro Harada, Finite simple groups with short chains of subgroups, J. Math. Soc. Japan 20 (1968), 655–672. MR 230811, DOI 10.2969/jmsj/02040655
David C. Hunt, A characterization of the finite simple group $M(22)$, J. Algebra 21 (1972), 103–112. MR 299673, DOI 10.1016/0021-8693(72)90038-5
David C. Hunt, A characterization of the finite simple group $M(23)$, J. Algebra 26 (1973), 431–439. MR 322046, DOI 10.1016/0021-8693(73)90004-5
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
Zvonimir Janko, A characterization of the finite simple group $\textrm {PSp}_{4}\,(3)$, Canadian J. Math. 19 (1967), 872–894. MR 214672, DOI 10.4153/CJM-1967-082-9

A characterization of the simple group

12

David Parrott, On Thompson’s simple group, J. Algebra 46 (1977), no. 2, 389–404. MR 447396, DOI 10.1016/0021-8693(77)90377-5
Kok-wee Phan, A characterization of the finite simple group $L_{4}\,(3)$, J. Austral. Math. Soc. 10 (1969), 51–76. MR 0251132
Kok-wee Phan, A characterization of the finite simple group $U_{4}\,(3)$, J. Austral. Math. Soc. 10 (1969), 77–94. MR 0258946
Michio Suzuki, On characterizations of linear groups. I, II, Trans. Amer. Math. Soc. 92 (1959), 191–219. MR 108536, DOI 10.1090/S0002-9947-1959-0108536-9
Helmut Wielandt, Beziehungen zwischen den Fixpunktzahlen von Automorphismengruppen einer eindlichen Gruppe, Math. Z 73 (1960), 146–158 (German). MR 0117286, DOI 10.1007/BF01162475
Jørn Børling Olsson, Odd-order extensions of some orthogonal groups, J. Algebra 28 (1974), 573–596. MR 427453, DOI 10.1016/0021-8693(74)90061-1
Ronald Solomon, Finite groups with Sylow $2$-subgroups of type $\Omega (7,\,q)$, $q\equiv \pm 3$ $(\textrm {mod}\ 8)$, J. Algebra 28 (1974), 174–181. MR 357601, DOI 10.1016/0021-8693(74)90030-1
J. H. Conway, Three lectures on exceptional groups, Finite simple groups (Proc. Instructional Conf., Oxford, 1969) Academic Press, London, 1971, pp. 215–247. MR 0338152
Dieter Held, A characterization of the alternating groups of degrees eight and nine, J. Algebra 7 (1967), 218–237. MR 218444, DOI 10.1016/0021-8693(67)90057-9
David C. Hunt, Character tables of certain finite simple groups, Bull. Austral. Math. Soc. 5 (1971), 1–42. MR 302753, DOI 10.1017/S0004972700046852
David C. Hunt, The character table of Fischer’s simple group, $M(23)$, Math. Comp. 28 (1974). MR 369494, DOI 10.1090/S0025-5718-1974-0369494-7
Ronald Solomon, Finite groups with Sylow $2$-subgroups of type ${\mathfrak {A}}_{12}$, J. Algebra 24 (1973), 346–378. MR 311762, DOI 10.1016/0021-8693(73)90144-0
R. G. Stanton, The Mathieu groups, Canad. J. Math. 3 (1951), 164–174. MR 40304, DOI 10.4153/cjm-1951-021-6
John G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383–437. MR 230809, DOI 10.1090/S0002-9904-1968-11953-6