Abelian invariants and a reduction theorem for the modular isomorphism problem (Q6054780)

Let \(p\) be a prime number, let \(G\) be a finite \(p\)-group and let \(k\) be a field of characteristic \(p\). We say that \(G\) satisfies the modular isomorphism problem if and only if for each group \(H\), the group algebras \(kG\) and \(kH\) are isomorphic if and only if the groups \(G\) and \(H\) are isomorphic. The classic statement of the \textit{modular isomorphism problem} asks whether this property holds for every finite \(p\)-group \(G\), under the additional hypothesis that \(k\) is the field with \(p\) elements. It already appeared in [\textit{R. Brauer}, Lect. Modern Math. 1, 133--175 (1963; Zbl 0124.26504)], and the first partial positive result goes back to [\textit{W. E. Deskins}, Duke Math. J. 23, 35--40 (1956; Zbl 0075.23905)]. Even though the statement of the modular isomorphism problem is now known to be false in general due to the example provided in [\textit{D. García-Lucas} et al., J. Reine Angew. Math. 783, 269--274 (2022; Zbl 1514.20019)], for odd primes it remains open. In the paper under review the authors provide the first known ``reduction theorem'' for the modular isomorphism problem. Here by reduction theorem we mean a result that allows to answer the mentioned problem over the class of all finite \(p\)-groups studying only a proper subclass. For them (see Theorem 4.1), this subclass is the one of finite \(p\)-groups without elementary abelian direct factors. A simplified version of this reduction is stated by the authors in Theorem A, which reads as follows: if \(G, H\) and \(E\) are finite \(p\)-groups and moreover \(E\) is elementary abelian, then \(k(G\times E)\cong k(H\times E)\) implies that \(kG\cong kH\). A group theoretical feature of \(G\) is said to be determined by its group algebra over \(k\) if for any group \(H\) such that \(kG \cong kH\), the group \(H\) has the same feature. The authors show in Theorem B that the isomorphism types of the following abelian subquotients of \(G\) are determined by the group algebra \(kG\): \begin{itemize} \item \(G/\gamma(G)\Omega_n(\mathrm{Z}(G))\), \item \(\gamma(G)\Omega_n(\mathrm{Z}(G))/\gamma(G)\), \item \(\mathrm{Z}(G)\cap \mho_n(G) \gamma(G) \), \item \(\mathrm{Z}(G)/\mathrm{Z}(G)\cap \mho_n(G)\gamma(G)\), \end{itemize} where \(\gamma(G)\) denotes the derived subgroup of \(G\), \(\mathrm{Z}(G)\) denotes the center of \(G\), and for a subset \(X\) of \(G\), one denotes \(\Omega_n(X)=\left\langle x\in X: x^{p^n}=1 \right\rangle \) and \(\mho_n(X)=\left\langle x^{p^n} : x\in X \right \rangle\). As an application of these new invariants, the authors show that modular isomorphism problem has positive answer for a \(2\)-group \(G\) provided that \(\mathrm{Z}(G)\) is cyclic and \(G/\mathrm{Z}(G)\) is dihedral. It should be mentioned that inspired by this paper the reviewer has generalized Theorem A in [``The modular isomorphism problem and abelian direct factors'', Mediterr. J. Math. 21, 18 (2024)] by dropping the ``elementary'' hypothesis when the field \(k\) has \(p\) elements, i.e., reducing the modular isomorphism problem to finite \(p\)-groups without abelian direct factors.