Bilinear and quadratic maps, and some \(p\)-groups of class 2 (Q1320194)

The author consider bilinear maps \(B:V \times V \to W\), where \(V,W\) are vector spaces over a field \(F\). An equivalence of \(B\) with another such map \(B_ 1 : V_ 1 \times V_ 1 \to W_ 1\) is a pair \((f,g)\) of linear isomorphisms \(f:V\to V_ 1\), \(g : W \to W_ 1\) such that \(g(B(x,y)) = B_ 1(f(x), f(y))\) for all \(x,y \in V\). The vector spaces of interest in this paper are primarily extension fields of the base field. Specifically, let \(E/F\) be a finite Galois extension , \(K\) an intermediate field, and \(\Phi\) the set of all \(F\)- monomorphisms of \(K\) into \(E\). For each \(\sigma \in \Phi\), one obtains bilinear maps \(B_ \sigma : K \times K \to E\) and \(B_ \sigma' : K \times K \to E\) defined by \(B_ \sigma (x,y) = x \sigma (y) - \sigma (x)y\) and \(B_ \sigma'(x,y) = x \sigma (y) + \sigma (x)y\) and a corresponding quadratic map \(Q_ \sigma : K \to E\) via \(Q_ \sigma (x) = x \sigma (x)\). In the case that \(K = E\), it is proved that \(B_ \tau\) is equivalent to \(B_ \sigma\) if and only if \(\tau\) is conjugate in \(\Phi\) to either \(\sigma\) or \(\sigma^{-1}\). Similar results are obtained for the maps \(B_ \sigma'\) and \(Q_ \sigma\). The connection with group theory is through certain groups of matrices corresponding to automorphisms of a finite field \(E\) of characteristic \(p\). For an automorphism \(\sigma\) of \(E\), let \(G_ \sigma\) be the group of all matrices of the form \(\left (\begin{smallmatrix} 1 & x & y \\ 0 & 1 & \sigma (x) \\ 0 & 0 & 1 \end{smallmatrix} \right)\). If \(\sigma \neq 1\), \(G_ \sigma\) is a \(p\)-group of nilpotency class 2. The above theorem on equivalence of bilinear maps is applied to prove that \(G_ \tau\) is isomorphic to \(G_ \sigma\) if and only if \(\tau = \sigma\) or \(\tau = \sigma^{-1}\). The result on equivalence of quadratic maps is further extended to a result on representations, leading to information on homomorphisms between groups of the form \(G_ \sigma\). Finally, automorphisms of the bilinear and quadratic maps are studied, and the structure of the automorphism group of \(G_ \sigma\) is determined, modulo central automorphisms.