= Odd unit group forces characteristic two
{title2=$|U(R)|\text{ odd},\ R\ne0\Longrightarrow\operatorname{char}R=2$}
In a nonzero unital <ring> with finite odd <unit group>, the <unit> $-1$ has order dividing two. By <Lagrange's theorem>, that order also divides $|U(R)|$, so $-1=1$ and the <characteristic of a ring> is two. Consequently $\overline a\mapsto a1_R$ defines a unital <ring homomorphism> $\mathbb F_2\to R$. This map still exists for the <zero ring>, whose characteristic is one under the least-positive-integer convention.
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