Character triples and relative defect zero characters

Abstract Given a character triple <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo>,</m:mo> <m:mi>θ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (G,N,\theta) , which means that 𝐺 is a finite group with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="normal">⊲</m:mi> <m:mo>⁢</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> N\vartriangleleft G and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>θ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>Irr</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>N</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \theta\in\operatorname{Irr}(N) is 𝐺-invariant, we introduce the notion of a 𝜋-quasi-extension of 𝜃 to 𝐺, where 𝜋 is the set of primes dividing the order of the cohomology element <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>θ</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> <m:mo>,</m:mo> <m:msup> <m:mi mathvariant="double-struck">C</m:mi> <m:mo>×</m:mo> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> [\theta]_{G/N}\in H^{2}(G/N,\mathbb{C}^{\times}) associated with the character triple, and then we establish the uniqueness of such an extension in the normalized case. As an application, we use the 𝜋-quasi-extension of 𝜃 to construct a bijection from the set of 𝜋-defect zero characters of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> G/N onto the set of relative 𝜋-defect zero characters of 𝐺 over 𝜃. Our results generalize the related theorems of M. Murai and of G. Navarro.