A class of finite <i>p</i> -groups and the normalized unit groups of group algebras (Q6600709)

Given a finite \(p\)-group \(G\) and field \(K\) a field of characteristic \(p\), a general problem of main interest is which group-theoretical structure of \(G\) is determined by its modular group algebra \(KG\). When something is determined, the second natural question is ``how to read it off \(KG?\)''. In this spirit a question was posed in [\textit{D. L. Johnson}, Proc. Am. Math. Soc. 68, 19--22 (1978; Zbl 0264.20020)] about the \(p\)-th power subgroups: ``is \(G^p = (1+ \omega(KG))^p \cap G\)?'' where \(\omega(KG)\) is the augmentation ideal of \(KG\). Note that \( 1+ \omega(KG)\) is a group as it equals \(V(KG)\), the group of units of augmentation one. More generally one can ask whether \(V(KG)^{p^{\ell}}\) is related to \(G^{p^{\ell}}\) and the same about their \(\Omega_{\ell}\) subgroups.\N\NThis article contributes to the understanding of above problems. More precisely in Theorem 1.1. the authors first classify all finite \(p\)-groups \(G\) with \(G' \cong C_p\) and which furthermore are central extensions of \(C_{p^n} \times C_{p^m}\) with \(C_p^k\) for some positive integers \(n,m,k\). The classification is subdivided in several cases depending on the isomorphism type of the center of \(G\). Next, in Theorem 1.2, the aforementioned questions are answered for the groups classified in Theorem 1.1 and with \(K = \mathbb{F}_p\). More precisely it is shown that: (i) \(G^p = V(\mathbb{F}_pG)^p \cap G\); (ii) \(V(\mathbb{F}_pG)^{p^{\ell}} = V(\mathbb{F}_pG^{p^{\ell}})\) and (iii) \(\Omega_{\ell}(V(\mathbb{F}_p G)) = 1 + \omega(G, \Omega_{\ell}(G))\) for \(\ell \geq 2\).