Chains of idempotents in endomorphism monoids
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CHAINS OF IDEMPOTENTS IN ENDOMORPHISM MONOIDS B.A.F. Wehrfritz
Received: 18 October 2012 / Accepted: 25 November 2012 / Published online: 9 October 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract If G is a group with finite Hirsch number and with its maximal locally finite normal subgroup satisfying the minimal condition on subgroups, e.g. if G is a finite extension of a torsion-free soluble group of finite rank, then there exists an integer k = k(G) such that for every subgroup H of G any chain of idempotents in the endomorphism monoid End(H ) of H has length at most k. Keywords Soluble group · Rank restrictions · Chains of idempotent endomorphisms Mathematics Subject Classification (2010) 20F16 · 20M20
1 Introduction In [5] for reasons given there we speculated that the endomorphism monoid End(G) of a (torsion-free)-by-finite soluble group G of finite rank should be embeddable into the multiplicative monoid of a matrix ring of finite degree over the rationals (or at least over the complex numbers). Theorem 2 of [5] shows that this does hold if G is torsion-free nilpotent. The paper [5] also contains further positive evidence (namely that periodic subsemigroups of such an End G are locally finite, the polycyclic case being due to Endimioni [1]). Here we present additional positive evidence. A group G has finite Hirsch number if it has a series of finite length whose factors are infinite cyclic or locally finite, the number h(G) of infinite cyclic factors in such a series being an invariant of G called the Hirsch number of G. A soluble group of finite rank clearly has finite Hirsch number. We prove the following. Theorem Let G be a group with finite Hirsch number and with its maximal locally finite normal subgroup τ (G) satisfying min, the minimal condition on subgroups. Then there exists
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B.A.F. Wehrfritz ( ) School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, England, UK e-mail: [email protected]
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B.A.F. WEHRFRITZ
an integer k = k(G) such that for every subgroup H of G any chain of idempotents in the endomorphism monoid End(H ) of H has length at most k. We present two proofs of this theorem. Both proofs are elementary. The first follows an obvious line of attack while the second is less direct. (Note that matrix rings do satisfy the property of the theorem, see Step 1 below.) Let S be a semigroup. If e and f are idempotents of S write e ≥ f if ef = f = f e. This is a (well known) partial order. Define idd(S) by idd(S) = ∅ if S has no idempotents and idd(S) = sup{r : there exist idempotents e0 > e1 > · · · > er in S}, so idd(S) is a non-negative integer or ∞ or ∅. If T is also a semigroup then idd(S × T ) = ∅ if either idd(S) or idd(T ) is ∅. Otherwise idd(S × T ) = idd(S) + idd(T ). If G is a group, then End(G) always contains 1 and 0, so idd(End(G)) ≥ 1 unless |G| = 1, when idd(End(G)) = 0.
2 Proof of the Theorem Step 1 If n is a positive integer and R a
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