Ultraproducts of quasirandom groups with small cosocles

Yang, Yilong

doi:10.1515/jgth-2016-0012

Cited by 5 publications

(8 citation statements)

References 13 publications

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“…In Section 2, we shall establish the proof of Theorem 1.3, using mostly elementary arguments combined with the main results on ultraproducts of finite simple groups from [Yan16]. We can break it down into the following subsections:…”

Section: Outline Of the Papermentioning

confidence: 99%

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Projective Limits and Ultraproducts of Nonabelian Finite Groups

Nahlus,

Yang

2021

Preprint

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Groups that can be approximated by finite groups have been the center of much research. This has led to the investigations of the subgroups of metric ultraproducts of finite groups. This paper attempts to study the dual problem: what are the quotients of ultraproducts of finite groups?Since an ultraproduct is an abstract quotient of the direct product, this also led to a more general question: what are the abstract quotients of profinite groups? Certain cases were already well-studied. For example, if we have a pro-solvable group, then it is well-known that any finite abstract quotients are still solvable.This paper studies the case of profinite groups from a surjective inverse system of non-solvable finite groups. We shall show that, for various classes of groups C, any finite abstract quotient of a pro-C group is still in C. Here C can be the class of finite semisimple groups, or the class of central products of quasisimple groups, or the class of products of finite almost simple groups and finite solvable groups, or the class of finite perfect groups with bounded commutator width.Furthermore, in the case of finite perfect groups with unbounded commutator width, we show that this is NOT the case. In fact, any finite group could be a quotient of an ultraproduct of finite perfect groups. We provide an explicit construction on how to achieve this for each finite group.

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Section: Outline Of the Papermentioning

confidence: 99%

“…By Proposition 2.3, we know that we can find an ultrafilter U such that N U ⊆ M . But by [Yan16], G/N U is either isomorphic to S i for some i, or it cannot have any non-trivial finite dimensional unitary representation. But G/M is a finite quotient of G/N U , which must have a non-trivial finite dimensional unitary representation.…”

Section: Semisimple Groupsmentioning

confidence: 99%

Projective Limits and Ultraproducts of Nonabelian Finite Groups

Nahlus,

Yang

2021

Preprint

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“…In general, an ultra‐quasirandom group might not be

d

‐quasirandom for some

d ⩾ 1

. In fact in , Yang provides an example of an ultra‐quasirandom group which is not even 2‐quasirandom. This example appears there as Example 1.7 and it is attributed to Pyber.…”

Section: Compactifications Of Ultraproductsmentioning

confidence: 99%

“…To finish this section, we see that Pyber's example (see [, Example 1.7]) yields the existence of an ultraproduct of finite groups where the Bohr compactification and the universal internal compactification do not agree. Example Let

p ⩾ 5

be a prime and let

{false(G_{n} false)}_{n \in double-struckN}

be an infinite sequence of finite groups

G_{n}

satisfying the property that

G_{n}

is a perfect group with an element

a_{n}

which cannot be written as the product of

n + 1

commutator elements, and that the only simple quotient of

G_{n}

{normalPSL}_{2} false(F_{p^{n}} false)

.…”

Section: Compactifications Of Ultraproductsmentioning

confidence: 99%

On compactifications and product‐free sets

Palacín

2019

J. London Math. Soc.

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A subset of a group is said to be product free if it does not contain three elements satisfying the equation xy=z. We give a negative answer to a question of Babai and Sós on the existence of large product‐free sets in finite groups by model theoretic means. This question was originally answered by Gowers. Furthermore, we give a natural and sufficient model theoretic condition for a group to have a large product‐free subset, as well as a model theoretic account of a result of Nikolov and Pyber on triple products.

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“…(In the special case of SO(3, R) Wis and L. C. Robertson [24] gave a transparent, nearly elementary proof). For more about vdW-groups see [50], [64], [74], and [94].…”

Section: Theorem 32 For Every Group G There Exists a Functionmentioning

confidence: 99%