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Transactions of the American Mathematical Society

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On $ 2$-groups with no normal abelian subgroups of rank $ 3$, and their occurrence as Sylow $ 2$-subgroups of finite simple groups


Author: Anne R. MacWilliams
Journal: Trans. Amer. Math. Soc. 150 (1970), 345-408
MSC: Primary 20.29
MathSciNet review: 0276324
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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 [Enhancements On Off] (What's this?)

  • [1] J. L. Alperin, Centralizers of abelian normal subgroups of 𝑝-groups, J. Algebra 1 (1964), 110–113. MR 0167528
  • [2] N. Blackburn, On a special class of $ p$-groups, Acta Math. 100 (1958), 45-92. MR 21 #1349.
  • [3] -, Generalizations of certain elementary theorems on $ p$-groups, Proc. London Math. Soc. (3) 11 (1961), 1-22. MR 23 #A208.
  • [4] R. Brauer, Some applications of the theory of blocks of characters of finite groups. II, J. Algebra 1 (1964), 307-334. MR 30 #4836.
  • [5] W. Feit, Characters of finite groups, Notes printed by Yale University, New Haven, Conn., 1965.
  • [6] W. Feit and J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029. MR 29 #3538.
  • [7] G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403-420. MR 34 #2681.
  • [8] G. Higman, Suzuki $ 2$-groups, Illinois J. Math. 7 (1963), 79-96. MR 26 #1365.
  • [9] B. Huppert, Endliche Gruppen. I, Die Grundlehren der math. Wissenschaften, Band 134, Springer-Verlag, Berlin and New York, 1967. MR 37 #302.
  • [10] J. Thompson, Non-solvable finite groups whose nonidentity solvable subgroups have solvable normalizers (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0276324-3
Keywords: $ p$-group, Sylow $ p$-subgroup, automorphism, rank, generators, conjugate, transfer homomorphism, fusion
Article copyright: © Copyright 1970 American Mathematical Society