During the work on groups of local chacteristic \(p\), [see Groups, combinatorics, geometry, Durham 2001, World Scientific. 155-192 (2003; Zbl 1031.20008) for an overview] due to U. Meierfrankenfeld, B. Stellmacher and the reviewer, the importance of a certain type of so-called \(2F\)-modules \(V\) came up. These are \(\text{GF}(p)\)-modules \(V\) for a group \(G\) such that for some \(p\)-subgroup \(1\neq A\) of \(G\) we have \(|V/C_V(A)|\leq|A|^2\). This generalizes the notion of \(F\)-modules, where \(|V:C_V(A)|\leq|A|\). In a series of papers [J. Algebra 257, No. 2, 348-372 (2002; Zbl 1017.20005); J. Algebra 307, No. 2, 643-676 (2007; Zbl 1115.20002); Finite groups 2003. Proc. Gainesville conf. finite groups, Berlin: Walter de Gruyter. 117-183 (2004; Zbl 1085.20004)], R. M. Guralnick, R. Lawther and G. Malle determine the irreducible \(2F\)-modules for the automorphism groups of the quasi-simple groups. In the work quoted above the module has additional properties. First \([V,A,A,A]= 0\), which is said to be cubic action. Furthermore, \[ [V, A]+C_V(A)=[\langle v\rangle,A]+C_V(A)\quad\text{for each }v\in V\setminus[V, A]+C_V(A).\tag{*} \] These two conditions, i.e., cubic and (*), the authors took as the definition of a nearly quadratic module. It really generalizes the quadratic modules, which have \([V,A,A]=0\) and \([V,A]\leq C_V(A)\).
In this paper, the authors prove many interesting facts about nearly quadratic modules. Among them the three main theorems. The first one is a reduction theorem to simple modules. If \(V\) is a semisimple module for \(G\) and \(G=\langle\mathcal Q\rangle\), \(\mathcal Q\) a set of nearly quadratic (but not quadratic) subgroups. Then we get a partition \(G=\bigtimes_{i\in I}G_i\), \(G_i=\langle\mathcal Q_i\rangle\), \(\mathcal Q_i\subseteq Q\), \(V=C_V(G)\oplus\bigoplus_{i\in I}[V,G_i]\), where the \([V,G_i]\) are simple \(G_i\)-modules.
The second theorem treats the situation that \(V\) is simple and \(F^*(G)\neq Z(G)K\), \(K\) a component, for which \(V\) is a simple \(K\)-module. There are thirteen cases, which cannot be stated here.
Finally, as an application of this theorem, the authors show that if \(V\) is a faithful module with \(A\) acting nearly quadratically on \(V\) and \(K\) be a component of \(G\) such that \([K,A]\nleq K\) and \(|A:C_A(K)|>2\), then \(p=2\), \(|A:N_A(K)|=2\) and \(K\cong L_n(2)\) or \(L_2(2^m)\). It is worthwhile mentioning that all results in this paper are proved without using the classification of the finite simple groups.