What is the automorphism group of the additive group of the dyadic rationals $\mathbb{Z}[\frac{1}{2}]$ ?
It is its own endomorphism ring, so its automorphism group is just its group of units (acting by multiplication). $\langle -1 \rangle \times \langle 2 \rangle \cong C_2 \times C_\infty$ – Jack Schmidt Jul 2 '12 at 16:37