On abelian quotients of primitive groups
HTML articles powered by AMS MathViewer
- by Michael Aschbacher and Robert M. Guralnick PDF
- Proc. Amer. Math. Soc. 107 (1989), 89-95 Request permission
Abstract:
It is shown that if $G$ is a primitive permutation group on a set of size $n$, then any abelian quotient of $G$ has order at most $n$. This was motivated by a question in Galois theory. The field theoretic interpretation of the result is that if $M/K$ is a minimal extension and $L/K$ is an abelian extension contained in the normal closure of $M$, then the degree of $L/K$ is at most the degree of $M/K$.References
-
M. S. Audu, Transitive permutation groups of prime-power order, Ph. D. Thesis, Oxford, 1983.
- M. Aschbacher and L. Scott, Maximal subgroups of finite groups, J. Algebra 92 (1985), no. 1, 44–80. MR 772471, DOI 10.1016/0021-8693(85)90145-0
- 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, DOI 10.1093/qmath/39.3.361
- 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
- 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, DOI 10.2307/2322773
Additional Information
- © Copyright 1989 American Mathematical Society
- 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