On (𝑛 + ½)-Engel groups

Abstract

Abstract Let n be a positive integer. We say that a group G is an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {(n+\frac{1}{2})} -Engel group if it satisfies the law <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mmultiscripts> <m:mi>y</m:mi> <m:mprescripts /> <m:mi>n</m:mi> <m:none /> </m:mmultiscripts> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo>]</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {[x,{}_{n}y,x]=1} . The variety of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {(n+\frac{1}{2})} -Engel groups lies between the varieties of n -Engel groups and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {(n+1)} -Engel groups. In this paper, we study these groups, and in particular, we prove that all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>4</m:mn> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {(4+\frac{1}{2})} -Engel <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>3</m:mn> <m:mo>}</m:mo> </m:mrow> </m:math> {\{2,3\}} -groups are locally nilpotent. We also show that if G is a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>4</m:mn> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {(4+\frac{1}{2})} -Engel p -group, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>5</m:mn> </m:mrow> </m:math> {p\geq 5} is a prime, then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> {G^{p}} is locally nilpotent.