Shifted addition on odd integers
= Shifted addition on odd integers
{title2=$a*b=a+b+1$}
The <odd integers> form an <abelian group> under the displayed operation. The <identity element> is $-1$ and the <inverse element> of $a$ is $-a-2$. The <bijection> $a\mapsto(a+1)/2$ is a <group isomorphism> to $(\mathbb Z,+)$, since it converts the shifted operation into addition. This illustrates how an unfamiliar identity and inverse can arise from ordinary addition by a change of coordinates.