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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Abelian $ {\rm f.p.f.}$ operator groups of type $ (p,\,p)$


Author: James W. Richards
Journal: Proc. Amer. Math. Soc. 29 (1971), 1-9
MSC: Primary 20.22
DOI: https://doi.org/10.1090/S0002-9939-1971-0274568-4
MathSciNet review: 0274568
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Abstract: A group A of automorphisms on a group G is said to be fixed-point-free, written f.p.f., if $ {C_G}(A) = \{ g \in G\vert{g^\alpha } = g$ for all $ \alpha \in A\} = I$. It has been shown by E. Shult that if A is an abelian f.p.f coprime group of automorphisms of order $ n = p_1^{{a_1}} \cdots p_k^{{a_k}}$ on a solvable group G, then the nilpotent length of G is bounded above by $ \psi (n) = \sum\nolimits_{i = 1}^{i = k} {{a_i}} $ unless $ \vert G\vert$ is divisible by primes q such that $ {q^s} + 1 = d$ where d divides the exponent e of A. F. Gross has removed the exceptional condition on the prime divisors of $ \vert G\vert$ when A is cyclic of order $ {p^a},p$, p an odd prime. In the case where A is noncyclic of order $ {p^2}$, the author has also removed the exceptional condition on the prime divisors of $ \vert G\vert$.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0274568-4
Keywords: Fixed-point-free, nilpotent length, Carter subgroup, Frattini subgroup, Frobenius group, absolutely irreducible representation, Fitting subgroup, hypercommutator subgroup, hypercenter, splitting field, descending nilpotent series
Article copyright: © Copyright 1971 American Mathematical Society

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