An upper bound on the Chebotarev invariant of a finite group

A subset { g 1 ,..., g d } of a finite group G invariably generates { g 1 x 1 , ... , g d x d } generates G for every choice of x i ∈ G . The Chebotarev invariant C ( G ) of G is the expected value of the random variable n that is minimal subject to the requirement that n randomly chosen elements of...

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Bibliographic Details
Published in:Israel journal of mathematics 2017-04, Vol.219 (1), p.449-467
Main Authors: Lucchini, Andrea, Tracey, Gareth
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Group Theory and Generalizations
Invariants
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
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Summary:A subset { g 1 ,..., g d } of a finite group G invariably generates { g 1 x 1 , ... , g d x d } generates G for every choice of x i ∈ G . The Chebotarev invariant C ( G ) of G is the expected value of the random variable n that is minimal subject to the requirement that n randomly chosen elements of G invariably generate G . The first author recently showed that C ( G ) ≤ β | G | for some absolute constant β . In this paper we show that, when G is soluble, then β is at most 5/3. We also show that this is best possible. Furthermore, we show that, in general, for each ε > 0 there exists a constant c ε such that C ( G ) ≤ ( 1 + ∈ ) | G | + c ∈ .
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-017-1507-x