Two applications of twisted wreath products to finite soluble groups

Author:
Trevor O. Hawkes

Journal:
Trans. Amer. Math. Soc. **214** (1975), 325-335

MSC:
Primary 20D10

MathSciNet review:
0379657

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Abstract: The group construction sometimes known as the twisted wreath product is used here to answer two questions in the theory of finite, soluble groups: first to show that an arbitrary finite, soluble group may be embedded as a subgroup of a group whose upper nilpotent series is a chief series; second to construct an *A*-group whose Carter subgroup is ``small'' relative to its nilpotent length.

**[1]**Roger Carter, Bernd Fischer, and Trevor Hawkes,*Extreme classes of finite soluble groups*, J. Algebra**9**(1968), 285–313. MR**0228581****[2]**Roger Carter and Trevor Hawkes,*The \cal𝐹-normalizers of a finite soluble group*, J. Algebra**5**(1967), 175–202. MR**0206089****[3]**E. C. Dade,*Carter subgroups and Fitting heights of finite solvable groups*, Illinois J. Math.**13**(1969), 449–514. MR**0255679****[4]**T. O. Hawkes,*An example in the theory of soluble groups*, Proc. Cambridge Philos. Soc.**67**(1970), 13–16. MR**0248225****[5]**T. O. Hawkes,*The family of Schunck classes as a lattice*, J. Algebra**39**(1976), no. 2, 527–550. MR**0407134****[6]**B. Huppert,*Endliche Gruppen. I*, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR**0224703****[7]**Gary M. Seitz,*Solvable groups having system normalizers of prime order*, Trans. Amer. Math. Soc.**183**(1973), 165–173. MR**0347970**, 10.1090/S0002-9947-1973-0347970-6**[8]**G. M. Seitz,*On system normalizers of prime order*, University of Oregon (preprint).**[9]**John G. Thompson,*Automorphisms of solvable groups*, J. Algebra**1**(1964), 259–267. MR**0173710**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0379657-X

Keywords:
Twisted wreath product,
nilpotent length,
Clifford theory,
primitive group,
Carter subgroup

Article copyright:
© Copyright 1975
American Mathematical Society