The generation of the augmentation ideal in profinite groups

Damian, Erika

doi:10.1007/s11856-011-0148-8

Cited by 5 publications

(5 citation statements)

References 20 publications

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“…Clearly, G is not finitely generated, as d(G) ≥ d(G * p i ,n i ) ≥ n i and the integers n i tend to infinity. Since G is soluble, by [11,Corollary 2.4] and the remark after it, it must not have type FP 1 .…”

Section: 1mentioning

confidence: 93%

“…It is interesting to compare this result with the equivalent condition for PFG given in [22,Theorem 11.1(3)], which is expressed entirely in terms of sizes of the crown-based powers of monolithic groups with non-abelian minimal normal subgroup which appear as quotients of G. On the other hand, the minimum number of generators for a profinite group can be determined by the crown-based powers of all monolithic primitive groups which appear as quotients of G (see [11,Lemma 4.2]).…”

Section: Type Fpmentioning

confidence: 99%

“…Our approach is rather direct and relies on elementary calculations. The method can be used to reprove results of Damian [11,Example 4.5], which rely on the machinery of [22]. We use our method to construct a UBERG group which is not finitely generated.…”

Section: Infinite Products Of Finite Groups With Ubergmentioning

confidence: 99%

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Counting irreducible modules for profinite groups

2022

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This article is concerned with the representation growth of profinite groups over finite fields. We investigate the structure of groups with uniformly bounded exponential representation growth (UBERG). Using crown-based powers we obtain some necessary and some sufficient conditions for groups to have UBERG. As an application we prove that the class of UBERG groups is closed under split extensions but fails to be closed under extensions in general. On the other hand, we show that the closely related probabilistic finiteness property PFP 1 is closed under extensions. In addition, we prove that profinite groups of type FP 1 with UBERG are always finitely generated and we characterise UBERG in the class of pronilpotent groups.Using infinite products of finite groups, we construct several examples with unexpected properties: (1) a UBERG group which cannot be finitely generated, (2) a group of type PFP ∞ which is not UBERG and not finitely generated, and (3) a finitely generated group of type PFP ∞ with superexponential subgroup growth.

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Section: 1mentioning

confidence: 93%

Section: Type Fpmentioning

confidence: 99%

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Counting irreducible modules for profinite groups

2022

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“…On the other hand, the following example shows that a group in L HẐF need not be countably based even if it is of type FP 1 . This example is adapted from [5,Example 2.6]; the approach is the same, but we construct groups which are not countably based.…”

Section: Pro-discrete R-modulesmentioning

confidence: 99%

On profinite groups of type FP∞

Cook

2016

Advances in Mathematics

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Suppose R is a profinite ring. We construct a large class of profinite groups L H R F, including all soluble profinite groups and profinite groups of finite cohomological dimension over R. We show that, if G ∈ L H R F is of type FP ∞ over R, then there is some n such that H n R (G, R G ) = 0, and deduce that torsionfree soluble pro-p groups of type FP ∞ over Z p have finite rank, thus answering the torsion-free case of a conjecture of Kropholler.

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“…The existence of a group of type FP 1 over Ẑ that is not finitely generated is shown in [5,Example 2.6]. (c) Let π be a set of primes.…”

Section: Profinite Group Homology and Cohomology Over Direct Systemsmentioning

confidence: 99%

Bieri-Eckmann Criteria for Profinite Groups

Cook¹

2014

Preprint

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In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FPn over a profinite ring R, analogous to the Bieri-Eckmann criteria for abstract groups. We use these to prove that the class of groups of type FPn is closed under extensions, quotients by subgroups of type FPn, proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP∞ over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FPn but not of type FP n+1 over Z Ĉ for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FPn fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.

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The generation of the augmentation ideal in profinite groups

Cited by 5 publications

References 20 publications

Counting irreducible modules for profinite groups

Counting irreducible modules for profinite groups

On profinite groups of type FP∞

Bieri-Eckmann Criteria for Profinite Groups

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