A characterization of finite soluble groups | Zendy

Nikolov Nikolay | Zendy; Segal Dan | Zendy

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A characterization of finite soluble groups

Author(s) -

Nikolov Nikolay,

Segal Dan

Publication year - 2007

Publication title -

bulletin of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 2.396

H-Index - 48

eISSN - 1469-2120

pISSN - 0024-6093

DOI - 10.1112/blms/bdl028

Subject(s) - mathematics , combinatorics , finite group , characterization (materials science) , prime (order theory) , order (exchange) , divisor (algebraic geometry) , group (periodic table) , cyclic group , discrete mathematics , chemistry , physics , abelian group , organic chemistry , finance , optics , economics

Let G be a finite soluble group of order m and let w ( x 1 , …, x n ) be a group word. Then the probability that w ( g 1 , …, g n ) = 1 (where ( g 1 , …, g n ) is a random n ‐tuple in G ) is at least p −( m − t ) , where p is the largest prime divisor of m and t is the number of distinct primes dividing m . This contrasts with the case of a non‐soluble group G , for which Abért has shown that the corresponding probability can take arbitrarily small positive values as n → ∞.

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