Norm formulas for finite groups and induction from elementary Abelian subgroups. (Q2509278)

Let \(G\) be a finite group acting by ring automorphisms on an arbitrary ring \(R\) with unit. For any subgroup \(U\) of \(G\) the `norm map' \(N_U\colon R\to R^U\) is defined by \(N_U(x)=\sum_{g\in U}gx\), where \(R^U\) denotes the subring of \(U\)-invariant elements in \(R\). In [\textit{E.~Aljadeff, Y.~Ginosar}, J. Algebra 179, No. 2, 599-606 (1996; Zbl 0844.16019), Theorem 1] it was shown that \(N_G\) is surjective onto \(R^G\) if and only if \(N_E\) is surjective onto \(R^E\) for every elementary Abelian subgroup \(E\) of \(G\). This generalizes \textit{L. G. Chouinard}'s theorem [J. Pure Appl. Algebra 7, 287-302 (1976; Zbl 0327.20020)] that asserts that a \(\mathbb{Z}[G]\)-module is projective if and only if for every elementary Abelian subgroup \(E\) of \(G\) it is projective as a \(\mathbb{Z}[E]\)-module. Aljadeff and Ginosar's result can be rephrased as follows: there is an element \(x_G\in R\) such that \(N_G(x_G)=1\) if and only if there is an element \(x_E\in R\) such that \(N_E(x_E)=1\) for every elementary Abelian subgroup \(E\) of \(G\). Using this statement, Shelah observed [see \textit{E.~Aljadeff, Y.~Ginosar}, loc. cit., Proposition 6] that there exist formulas expressing \(x_G\) polynomially over \(\mathbb{Z}\) in terms of the elements \(x_E\) and the elements of \(G\). The authors showed [in \textit{E.~Aljadeff, C.~Kassel}, Isr. J. Math. 129, 99-108 (2002; Zbl 1026.16020)] how to obtain formulas for all Abelian groups. It should also be noted that the first-named author obtained formulas for arbitrary groups acting on commutative rings (see [\textit{E.~Aljadeff}, Isr. J. Math. 86, No. 1-3, 221-232 (1994; Zbl 0830.13004)]). In this paper the authors present a general and systematic approach to the problem of finding explicit formulas for norm one elements in arbitrary noncommutative rings. Their approach consists in breaking the problem into three tasks. In the first task, a presentation of \(G\) is converted into a system of equations in indeterminates \(b(\sigma)\), one for each generator \(\sigma\) in the presentation. The second task involves finding solutions in a ring \(R\) to these equations; it follows from homological reasons that the system must have a solution, but to perform the task it is necessary that the solutions be explicit polynomials in the given data. The third task uses homological algebra to convert the solutions to the desired formula. They show how to solve explicitly Task 1 and 3. For Task 2 they do not give a general solution, but they provide solutions to the system of equations in some important cases (namely, dihedral and quaternionic 2-groups). They also show that the problem for a general group \(G\) can be reduced to a smaller class of groups, namely the class of extraspecial and almost extraspecial \(p\)-groups that are subquotiens of \(G\).