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

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), no. 1, 44–80. MR**772471**, 10.1016/0021-8693(85)90145-0**[KN]**L. G. Kovács and M. F. Newman,*Generating transitive permutation groups*, Quart. J. Math. Oxford Ser. (2)**39**(1988), no. 155, 361–372. MR**957277**, 10.1093/qmath/39.3.361**[KP]**L. G. Kovács and Cheryl E. Praeger,*Finite permutation groups with large abelian quotients*, Pacific J. Math.**136**(1989), no. 2, 283–292. MR**978615****[I]**David G. Cantor and I. M. Isaacs,*Problems and Solutions: Solutions of Advanced Problems: 6523*, Amer. Math. Monthly**95**(1988), no. 6, 561–562. MR**1541339**, 10.2307/2322773

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

Article copyright:
© Copyright 1989
American Mathematical Society