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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ p$-solvable groups with few automorphism classes of subgroups of order $ p$


Author: Fletcher Gross
Journal: Proc. Amer. Math. Soc. 30 (1971), 437-444
MSC: Primary 20.40
MathSciNet review: 0286887
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Abstract: This paper investigates the relationship between the p-length, $ {l_p}(G)$, of the finite p-solvable group G and the number, $ {a_p}(G)$, of orbits in which the subgroups of order p are permuted by the automorphism group of G. If $ p > 2$ and $ {a_p}(G) \leqq 2$, it is shown that $ {l_p}(G) \leqq {a_p}(G)$. If $ p = 2$ and $ {a_2}(G) = 1$, it is proved that either $ {l_p}(G) \leqq {a_p}(G)$ or $ G/{O_{2'}}(G)$ is a specific group of order 48.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0286887-6
PII: S 0002-9939(1971)0286887-6
Keywords: p-solvable groups, p-length, automorphism classes, p-subgroups
Article copyright: © Copyright 1971 American Mathematical Society