[1] The Alfvén-cyclotron anisotropy instability is driven by a sufficiently large proton temperature anisotropy T ? /T k > 1 where ? and k denote directions perpendicular and parallel, respectively, to the background magnetic field B o . Here kinetic linear theory for a magnetized, homogeneous, collisionless plasma is used to study this instability at propagation parallel to B o in the presence of a relatively tenuous, relatively cool, isotropic, singly ionized helium component. A sufficiently dense helium component splits the Alfvén-cyclotron instability into two branches: a proton cyclotron branch at frequencies above the helium cyclotron frequency but below the proton cyclotron frequency, and a helium-ion cyclotron branch at frequencies less than the helium ion cyclotron frequency. If the helium ions are much cooler than the protons and are sufficiently dense, the helium-ion cyclotron branch can become unstable at wavelengths considerably shorter than the unstable waves of the proton cyclotron branch, favoring excitation of enhanced fluctuations which resonate with geomagnetically trapped electrons of energies between 500 keV and 2 MeV.Citation: Gary, S. P., K. Liu, and L. Chen (2012), Alfvén-cyclotron instability with singly ionized helium: Linear theory,