Groups that are almost homogeneous
A group \(G\) is called (almost) homogeneous if the group \(\Aut G\) has at most (3 orbits) 2 orbits on \(G\). It is easy to see that the homogeneous abelian groups are exactly the additive groups of vector spaces. A commutative field \(F\) is called \(q\)-suited if the cyclotomic polynomial \(\Phi_q=(x^q-1)/(x-1)\) has \(q-1\) different roots in \(F\), and \(\Aut F\) acts transitively on the set of these roots. If \(F\) is a field, then \(\Delta(q,F,d)\) denotes the semidirect product of the additive group \(U\) of a \(d\)-dimensional vector space over \(F\) and \(\Omega=\{f\in F\mid f^q=1\}\) acting on \(V\) by scalar multiplication. Let \(F\) be a commutative field, let \(V\) and \(Z\) be vector spaces over \(F\), and consider a symplectic bilinear map \(\beta\colon V\times V\to Z\) (that is \((v,v)=0\) for each \(v\in V\)). Let \(GH(V,Z,\beta)\) be the set \(V\times Z\) endowed with the operation \((v,y)\circ(w,z)=(v+w,y+(v,w)+z)\). Then \(GH(V,Z,\beta)\) is a group and the operation \(\circ\) coincides with addition on each one-dimensional subspace of \(V\times Z\). The group \(GH(V,Z,\beta)\) is called a generalized Heisenberg group. Let \(G\) be an almost homogeneous group. Then (a) If the group \(G\) is abelian then it is the additive group either of a vector space or of a free module over the local ring \(Z(p^2)\) for some prime \(p\). (b) If the group \(G\) is a solvable torsion group of composite exponent then it is isomorphic to one of the groups \(\Delta(q,F,d)\) for some \(q\)-suited field \(F\). (c) If the group \(G\) is nilpotent of exponent \(p\) then it is a generalized Heisenberg group. (d) If the group \(G\) is nilpotent of exponent \(p^2\) then either it is abelian, or \(p=2\) and \(G\) is special with \(G/Z(G)\) isomorphic to \(Z(G)\).
scientific article; zbMATH DE number 1098484