Abstract
There are two important properties of groups that are stronger than commutativity: they are solvability and nilpotence. Solvable1 groups are obtained by forming successive extensions of abelian groups; nilpotent groups lie midway between abelian and solvable groups.
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Notes
- 1.
British English speakers often say “soluble” instead of “solvable”.
- 2.
But see ex. 20, iii).
- 3.
M. Kargapolov, Iou. Merzliakov, Theorem 16.3.2
- 4.
Definition 2.19.
- 5.
See ex. 31 of Chapter 3.
- 6.
V is the initial of Verlagerung, the German term for transfer.
- 7.
Cf. Lemma 3.8.
- 8.
Alperin J.L.: Sylow Intersections and Fusion. J. Algebra 6, 222–241 (1967), and Glauberman G.: Global and Local Properties of Finite Groups. In: Powell M.B., Higman G. (eds.), Finite Simple groups. Academic Press, London and New York (1971).
- 9.
This fact characterizes p-nilpotent groups (Huppert, p. 432).
- 10.
The converse also holds: if a group has a p-complement for all p, then it is solvable.
- 11.
The original proof makes use of the Frobenius theorem (Theorem 6.21).
- 12.
This proposition is the content of a celebrated theorem of W. Feit and J.G. Thompson.
- 13.
The result also holds without the hypothesis of solvability (Glauberman).
- 14.
This fact characterizes finite solvable groups (J.G. Thompson).
- 15.
We recall that an equation is solvable by radicals if and only if its Galois group is a solvable group (Remark 2 of 3.68).
References
Alperin J.L.: Sylow Intersections and Fusion. J. Algebra 6, 222–241 (1967), and Glauberman G.: Global and Local Properties of Finite Groups. In: Powell M.B., Higman G. (eds.), Finite Simple groups. Academic Press, London and New York (1971).
This fact characterizes p-nilpotent groups (Huppert, p. 432).
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Machì, A. (2012). Nilpotent Groups and Solvable Groups. In: Groups. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2421-2_5
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DOI: https://doi.org/10.1007/978-88-470-2421-2_5
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