= Solution
The sum of two <odd integers> plus one is odd, so the operation is closed. Both associations of a triple give $a+b+c+2$, proving <associativity>. Its <identity element> is $-1$, and the <inverse element> of $a$ is $-a-2$, also odd, since $a+(-a-2)+1=-1$. Thus it is an <abelian group>.
More explicitly, <shifted addition on odd integers> is ordinary integer addition in different coordinates:
$$
\boxed{F(a)=\frac{a+1}{2},\qquad F(a*b)=F(a)+F(b).}
$$
This <bijection> is a <group isomorphism> to $(\mathbb Z,+)$.
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