On $2$-groups with no normal abelian subgroups of rank $3$, and their occurrence as Sylow $2$-subgroups of finite simple groups
HTML articles powered by AMS MathViewer
- by Anne R. MacWilliams PDF
- Trans. Amer. Math. Soc. 150 (1970), 345-408 Request permission
Abstract:
We prove that in a finite $2$-group with no normal Abelian subgroup of rank $\geqq 3$, every subgroup can be generated by four elements. This result is then used to determine which $2$-groups $T$ with no normal Abelian subgroup of rank $\geqq 3$ can occur as ${S_2}$’s of finite simple groups $G$, under certain assumptions on the embedding of $T$ in $G$.References
- J. L. Alperin, Centralizers of abelian normal subgroups of $p$-groups, J. Algebra 1 (1964), 110–113. MR 167528, DOI 10.1016/0021-8693(64)90027-4 N. Blackburn, On a special class of $p$-groups, Acta Math. 100 (1958), 45-92. MR 21 #1349. —, Generalizations of certain elementary theorems on $p$-groups, Proc. London Math. Soc. (3) 11 (1961), 1-22. MR 23 #A208. R. Brauer, Some applications of the theory of blocks of characters of finite groups. II, J. Algebra 1 (1964), 307-334. MR 30 #4836. W. Feit, Characters of finite groups, Notes printed by Yale University, New Haven, Conn., 1965. W. Feit and J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029. MR 29 #3538. G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403-420. MR 34 #2681. G. Higman, Suzuki $2$-groups, Illinois J. Math. 7 (1963), 79-96. MR 26 #1365. B. Huppert, Endliche Gruppen. I, Die Grundlehren der math. Wissenschaften, Band 134, Springer-Verlag, Berlin and New York, 1967. MR 37 #302. J. Thompson, Non-solvable finite groups whose nonidentity solvable subgroups have solvable normalizers (to appear).
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 150 (1970), 345-408
- MSC: Primary 20.29
- DOI: https://doi.org/10.1090/S0002-9947-1970-0276324-3
- MathSciNet review: 0276324