Topological structure of a formal group on twice the 2-adic integers ($F(2{\mathbb{Z}}_2)\cong {\mathbb{Z}}_2\text{or}\mathbb{Z}/2\mathbb{Z}\times {\mathbb{Z}}_2$)

For a one-dimensional commutative integral formal group law, the formal logarithm identifies $F(4{\mathbb{Z}}_2)$ with $4{\mathbb{Z}}_2$. This is an index-two subgroup of $F(2{\mathbb{Z}}_2)$. Choose a topological generator $h$ of the subgroup and an element $v$ outside it, and write $2v=ch$. If $c$ is odd, $v$ generates the whole group. If $c$ is even, $v-(c/2)h$ has order two and splits off a cyclic factor. The formal additive group and formal multiplicative group realize the two cases.