Stabilisers of injective modules over nilpotent groups

Brookes, C. J. B.; Cheng, Kai N.; Leong, Yu K.

doi:10.1515/9783110848397-021

Cited by 4 publications

(12 citation statements)

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“…The next result answers Questions 1 and 2 of the first author in [1]. If H is a finitely generated torsion-free nilpotent group, then Aut H can be embedded as an arithmetic subgroup of GL(n, Q).…”

Section: Theorem a Let G Be A Torsion-free Connected Polycyclic Groumentioning

confidence: 72%

Some infinite soluble groups, their modules, and the arithmeticity of associated automorphism groups

Brookes¹,

Groves

2005

Journal of Algebra

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Section: Theorem a Let G Be A Torsion-free Connected Polycyclic Groumentioning

confidence: 72%

Some infinite soluble groups, their modules, and the arithmeticity of associated automorphism groups

Brookes¹,

Groves

2005

Journal of Algebra

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“…We say that p stabilises M if E(M) and E(Mp) = E(M)p are fctf-isomorphic. A further discussion of the necessary ideas can be found in [1].…”

Section: Other Free Nilpotent Groupsmentioning

confidence: 99%

“…Then we can suppose that M has a nonzero kH'/P.kH'-torsion submodule M x which is finitely generated as kH -module. Let E(M\) denote the injective hull of M\ and let W be an indecomposable injective it//'-submodule of E(M\) which is minimal in the sense of [1]. We aim to show that E(Mi) has a non-zero &H-submodule N which is induced from H'.…”

Section: Proposition 3 Suppose That H Is a Free Nilpotent Group And mentioning

confidence: 99%

“…Because a module is essential in its injective hull, it will follow that N n Mi is non-zero. But, it then follows from (for example) the Lemma of [1] that N n M\ contains a non-zero submodule induced from H'. This contradiction will then complete the proof.…”

Section: Proposition 3 Suppose That H Is a Free Nilpotent Group And mentioning

confidence: 99%

“…From the discussion preceding Theorem 2 of [1], we can assume that V is the injective hull of a finitely generated impervious &L-module Vj. We can now apply [1, Theorem 2] to deduce that V x .kH= V x .kS® kS kH where 5 is the stabiliser in N H {L) of Vj and so of V. Observe that if 5 < H' then Vj .kH is also induced from H' and the proof is complete.…”

Section: Proposition 3 Suppose That H Is a Free Nilpotent Group And mentioning

confidence: 99%

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Modules over infinite nilpotent groups

Groves¹

2001

J. Aust. Math. Soc.

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The paper discusses modules over free nilpotent groups and demonstrates that faithful modules are more restricted than might appear at first glance. Some discussion is also made of applying the techniques more generally.2000 Mathematics subject classification: primary 20F18, 20F18.The aim of this note is to discuss the nature of modules over finitely generated nilpotent groups. More particularly, we shall state some results for finitely generated faithful modules over free nilpotent groups and then briefly conjecture how these ideas might apply more generally. The techniques involve the application of results of Brookes [3] and of Brookes and the author [4,5,6].In contrast to the situation for modules over free abelian groups, it is not clear how to write down a wide range of faithful modules over an infinite nilpotent group. In order to avoid the complicated detail that comes with greater generality, we will restrict our consideration to faithful modules over free nilpotent groups of finite rank. Let H denote a free nilpotent group of finite rank with centre £ (H) and let k be a field. How can we construct a faithful module for kHlWe clearly have free modules as well as, possibly, other torsion-free modules. We also have some straightforward quotients of these. If P is a prime ideal of k£(H) satisfying (P + 1) D £(//) = 1 then, by a theorem of Zalesskii (see [9, Corollary 11.4.6]), kH/P.kH is a faithful module for H. Alternatively, we can regard such modules as being induced from modules over the centre of H (where it

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Finitely Presented Metanilpotent Groups

Groves

Wilson

1994

Journal of the London Mathematical Society

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FINITELY PRESENTED METANILPOTENT GROUPS 103 prime exponent. If M has exponent zero, then, being critical, it must be Z-torsionfree. It follows from [18, Theorem F] that M has a quotient of finite prime exponent and Krull dimension m-1. Replace M by this quotient. So M has prime exponent and Krull dimension m -rj, where rj is zero if G' has finite exponent and at most 1 otherwise.Since M corresponds to a quotient of G which is weakly finitely presented, we may apply Corollary 2.8 and conclude that By Lemma 4.3 we can find a subgroup H 1 of H with h(/f x ) = m such that M is finitely generated as Z// r module.

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Stabilisers of injective modules over nilpotent groups

Cited by 4 publications

References 0 publications

Some infinite soluble groups, their modules, and the arithmeticity of associated automorphism groups

Some infinite soluble groups, their modules, and the arithmeticity of associated automorphism groups

Modules over infinite nilpotent groups

Finitely Presented Metanilpotent Groups

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