On generating properties of the weak commutativity of <i>p</i> -groups, <i>p</i> odd

Abstract In the present paper, we examine the generating properties of Sidki’s weak commutativity group. More precisely, if G and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:mi>φ</m:mi> </m:msup> </m:math> {G^{\varphi}} are two isomorphic groups, the weak commutativity group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>χ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\chi(G)} is the group generated by G and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:mi>φ</m:mi> </m:msup> </m:math> {G^{\varphi}} subject to the relations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>g</m:mi> <m:mi>φ</m:mi> </m:msup> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {[g,g^{\varphi}]=1} for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> {g\in G} . Here we provide bounds for the number of generators of some subgroups of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>χ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\chi(G)} when G is a p -group of odd order and either G is powerful or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {D(G)} is abelian.