On abelian quotients of primitive groups

Authors:
Michael Aschbacher and Robert M. Guralnick

Journal:
Proc. Amer. Math. Soc. **107** (1989), 89-95

MSC:
Primary 20B05; Secondary 12F05, 20B25, 20B35

DOI:
https://doi.org/10.1090/S0002-9939-1989-0982398-4

MathSciNet review:
982398

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Abstract: It is shown that if is a primitive permutation group on a set of size , then any abelian quotient of has order at most . This was motivated by a question in Galois theory. The field theoretic interpretation of the result is that if is a minimal extension and is an abelian extension contained in the normal closure of , then the degree of is at most the degree of .

**[A]**M. S. Audu,*Transitive permutation groups of prime-power order*, Ph. D. Thesis, Oxford, 1983.**[AS]**M. Aschbacher and L. Scott,*Maximal subgroups of finite groups*, J. Algebra**92**(1985), 44-80. MR**772471 (86m:20029)****[KN]**L. G. Kovacs and M. F. Newman,*Generating transitive permutation groups*, Quart J. Math Oxford (2)**39**(1988), 361-372. MR**957277 (89i:20008)****[KP]**L. Kovacs and C. Praeger,*Finite permutation groups with large abelian quotients*, Pacific J. Math.,**136**(1989), 283-292. MR**978615 (89m:20001)****[I]**I. M. Isaacs,*Solution of problem*6523, Amer. Math. Monthly**95**(1988), 561-562. MR**1541339**

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DOI:
https://doi.org/10.1090/S0002-9939-1989-0982398-4

Article copyright:
© Copyright 1989
American Mathematical Society