Almost independence and irreducibility in simple finite and algebraic groups

Aner Shalev

doi:10.1016/j.jalgebra.2017.03.014

Almost independence and irreducibility in simple finite and algebraic groups

Aner Shalev

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study intersections of the form g₁C₁∩g₂C₂, where C_i are conjugacy classes of arbitrary finite simple groups and g_i are group elements. We show that, generically, |g₁C₁∩g₂C₂|∼|C₁||C₂|/|G|, which means that the events g₁C₁,g₂C₂ are almost independent in G. We also discuss the dimension and the irreducibility of such intersections in simple algebraic groups, and expose the anomaly of SL₂. This work is motivated by recent questions of Hrushovski.

Original language	American English
Pages (from-to)	375-389
Number of pages	15
Journal	Journal of Algebra
Volume	500
DOIs	https://doi.org/10.1016/j.jalgebra.2017.03.014
State	Published - 15 Apr 2018

Bibliographical note

Funding Information:
The author acknowledges the support of an Israel Science Foundation grant 1117/13, and of the Vinik Chair of Mathematics which he holds.

Publisher Copyright:
© 2017 Elsevier Inc.

Keywords

Zelmanov issue

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10.1016/j.jalgebra.2017.03.014

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title = "Almost independence and irreducibility in simple finite and algebraic groups",

abstract = "We study intersections of the form g1C1∩g2C2, where Ci are conjugacy classes of arbitrary finite simple groups and gi are group elements. We show that, generically, |g1C1∩g2C2|∼|C1||C2|/|G|, which means that the events g1C1,g2C2 are almost independent in G. We also discuss the dimension and the irreducibility of such intersections in simple algebraic groups, and expose the anomaly of SL2. This work is motivated by recent questions of Hrushovski.",

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N2 - We study intersections of the form g1C1∩g2C2, where Ci are conjugacy classes of arbitrary finite simple groups and gi are group elements. We show that, generically, |g1C1∩g2C2|∼|C1||C2|/|G|, which means that the events g1C1,g2C2 are almost independent in G. We also discuss the dimension and the irreducibility of such intersections in simple algebraic groups, and expose the anomaly of SL2. This work is motivated by recent questions of Hrushovski.

AB - We study intersections of the form g1C1∩g2C2, where Ci are conjugacy classes of arbitrary finite simple groups and gi are group elements. We show that, generically, |g1C1∩g2C2|∼|C1||C2|/|G|, which means that the events g1C1,g2C2 are almost independent in G. We also discuss the dimension and the irreducibility of such intersections in simple algebraic groups, and expose the anomaly of SL2. This work is motivated by recent questions of Hrushovski.

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DO - 10.1016/j.jalgebra.2017.03.014

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SN - 0021-8693

VL - 500

SP - 375

EP - 389

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Almost independence and irreducibility in simple finite and algebraic groups

Abstract

Bibliographical note

Keywords

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