On relative -normalizers in finite solvable groups

Author:
Patrick D’Arcy

Journal:
Proc. Amer. Math. Soc. **42** (1974), 381-384

MSC:
Primary 20D10

MathSciNet review:
0332956

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Abstract: In the theory of saturated formations of finite solvable groups, certain subgroups, called -reducers, have been considered. In this note it is pointed out that the work of an -reducer *R* is done by certain subsets of *R*, subsets that can be considered generalizations of P. Hall's relative system normalizers to an arbitrary subgroup of a given finite solvable group.

**[1]**Roger Carter and Trevor Hawkes,*The \cal𝐹-normalizers of a finite soluble group*, J. Algebra**5**(1967), 175–202. MR**0206089****[2]**Patrick D’Arcy,*𝔉-abnormality and the theory of finite solvable groups*, J. Algebra**28**(1974), 342–361. MR**0367051****[3]**C. J. Graddon,*𝔉-reducers in finite soluble groups*, J. Algebra**18**(1971), 574–587. MR**0280591****[4]**C. J. Graddon,*The relation between 𝔉-reducers and 𝔉-subnormalizers in finite soluble groups*, J. London Math. Soc. (2)**4**(1971), 51–61. MR**0292937****[5]**P. Hall,*On the system normalizers of a soluble group*, Proc. London Math. Soc.**43**(1937), 507-528.**[6]**B. Huppert,*Endliche Gruppen. I*, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR**0224703**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1974-0332956-4

Keywords:
Finite solvable group,
locally defined formation,
-reducer,
relative -normalizer

Article copyright:
© Copyright 1974
American Mathematical Society