Solvable groups having system normalizers of prime order
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- by Gary M. Seitz PDF
- Trans. Amer. Math. Soc. 183 (1973), 165-173 Request permission
Abstract:
Let G be a solvable group having system normalizer D of prime order. If G has all Sylow groups abelian then we prove that $l(G) = l({C_G}(D)) + 2$, provided $l(G) \geq 3$ (here $l(H)$ denotes the nilpotent length of the solvable group H). We conjecture that the above result is true without the condition on abelian Sylow subgroups. Other special cases of the conjecture are handled.References
- Roger Carter and Trevor Hawkes, The ${\cal F}$-normalizers of a finite soluble group, J. Algebra 5 (1967), 175β202. MR 206089, DOI 10.1016/0021-8693(67)90034-8
- Gary M. Seitz and C. R. B. Wright, On complements of ${\mathfrak {F}}$-residuals in finite solvable groups, Arch. Math. (Basel) 21 (1970), 139β150. MR 271234, DOI 10.1007/BF01220895
- John G. Thompson, Fixed points of $p$-groups acting on $p$-groups, Math. Z. 86 (1964), 12β13. MR 168653, DOI 10.1007/BF01111272
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 183 (1973), 165-173
- MSC: Primary 20D10
- DOI: https://doi.org/10.1090/S0002-9947-1973-0347970-6
- MathSciNet review: 0347970