A kind of imperfect field

Question

Facts:

(1) A: finite set of combinatorial nature |A|=n,

(2) (A,+,0) abelian group (isomorphic with Zn),

(3) (A,*,0) non-abelian group (with same identity as + op),

(4) * distributes (left & right) over +

Question: What name could this 2-op structure have? (If any)

Thanks,

Peter

Answer 1

No such structure exists unless $n=1$. Indeed, let $A$ be any abelian group with operation $+$ and identity element $0$, and suppose $*$ is a binary operation on $A$ which also makes $A$ a group and left-distributes over $+$. For any $a\in A$ we then have $$a*0=a*(0+0)=(a*0)+(a*0)$$ and subtracting $a*0$ gives $a*0=0.$ Now let $b\in A$ be arbitrary and take $a$ to be $b*0^{-1}$ (where $0^{-1}$ is the inverse of $0$ with respect to $0$). We then have $$0=a*0=(b*0^{-1})*0=b.$$ Thus every element of $A$ is equal to $0$, so $A$ has only one element.