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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 → ∞.