Why is Diffie-Hellman defined on a cyclic group? Doesn't it work for any commutative operation which the inverse is hard to find?
Say Alice and Bob agree in a public prime $c$ and both choose a secret prime $a$ respectively $b$
Alice sends $ac$ to Bob and Bob $bc$ to Alice.
Alice then multiplies $a$ with bobs message $bc$ yielding $abc$ Bob then multiplies $b$ with Aice's message $ac$ yield $bac$
which are the same due to commutativity and associativity. Hence they now share a common secret $abc$.
It is hard for Eve to factorize $ac$ and $bc$ into its original primes $a,b,c$ and Eve hasn't got enough information to construct $abc$ so why isn't this a valid Diffie-Hellman key-exchange?