6533b850fe1ef96bd12a8433

RESEARCH PRODUCT

Soluble groups with their centralizer factor groups of bounded rank

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Abstract For a group class X , a group G is said to be a C X -group if the factor group G / C G ( g G ) ∈ X for all g ∈ G , where C G ( g G ) is the centralizer in G of the normal closure of g in G . For the class F f of groups of finite order less than or equal to f , a classical result of B.H. Neumann [Groups with finite classes of conjugate elements, Proc. London Math. Soc. 1 (1951) 178–187] states that if G ∈ C F f , the commutator group G ′ belongs to F f ′ for some f ′ depending only on f . We prove that a similar result holds for the class S r ( d ) , the class of soluble groups of derived length at most d which have Prufer rank at most r . Namely, if G ∈ C S r ( d ) , then G ′ ∈ S d r ′ ( d ) for some r ′ depending only on r . Moreover, if G ∈ C ( S r ( d ) F f ) , then G ′ ∈ S r ′ ( d + 3 ) F f ′ for some r ′ and f ′ depending only on r , d and f .