Archivo por meses: agosto 2018

Paper «A note on normal complements for finite groups» published in Bull. Austral. Math. Soc.

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Ning Su, Adolfo Ballester-Bolinches, Hangyang Meng.

A note on normal complements for finite groups

Bull. Austral. Math. Soc., 98 (1):109-112, 2018

doi:10.1017/S0004972718000151

Abstract

Assume that G is a finite group and H is a 2-nilpotent Sylow tower Hall subgroup of G such that if x and y are G-conjugate elements of H ∩ G0 of prime order or order 4, then x and y are H-conjugate. We prove hat there exists a normal subgroup N of G such that G = HN and H ∩ N = 1.

2010 Mathematics Subject Classification: primary 20D20; secondary 20D10

Keywords: finite group, conjugation, Hall subgroup, normal complement.

Paper «When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?» published in Log. J. IGPL

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J. Climent Vidal, E. Cosme Llópez.
When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?
Log. J. IGPL, 26(4):381-407, 2018.

doi: 10.1093/jigpal/jzy005

Abstract:

For a set of sorts S and an S-sorted signature Σ we prove that a profinite Σ-algebra, i.e. a projective limit of a projective system of finite Σ-algebras, is a retract of an ultraproduct of finite Σ-algebras if the family consisting of the finite Σ-algebras underlying the projective system is with constant support. In addition, we provide a categorial rendering of the above result. Specifically, after obtaining a category where the objects are the pairs formed by a nonempty upward directed preordered set and by an ultrafilter containing the filter of the final sections of it, we show that there exists a functor from the just mentioned category whose object mapping assigns to an object a natural transformation which is a retraction.

2020 Mathematics Subject Classification: 03C20, 08A68, 18A30.

Keywords: support of a many-sorted set, family of many-sorted algebras with constant support, profinite, retract, projective limit, inductive limit, ultraproduct.