Which properties of a group are used in the steps of Diffie Hellman? I’m trying to understand which properties of a group are used in DHKE at each step. 

For example, Alice and Bob’s public keys appear to only use the closure property of a group and maybe identity  (e.g. $k_{pubA}$ = $A^{k_{prA}}$ (mod p)?

When creating the shared key Alice and Bob appear to also use the associative property of a group $k_{AB}$ = $B^{k_{prA}}$ (mod p)?

So to perform both main steps of DHKE the multiplicative inverse property does not seem to be used at all?
I’m trying to understand which properties of a group are used in DHKE at each step. 


Actually, you can implement a DH-style operation in any semigroup; you need closure, and you need associativity (so $A^3 = A\times A \times A = (A \times A) \times A = A \times (A \times A)$ is well defined), but other than that, you really don't need anything.  You don't need an identity, you don't need the semigroup to be abelian (although the sub-semigroup generated by a single element will always be abelian), it doesn't have to be finite (although infinite semigroups would cause practical problems during implementation) and you don't need inverses (which, if you don't have an identity, aren't well-defined anyways).

We typically don't talk about doing DH in a semigroup mostly because (AFAIK) no one has found a semigroup (that's not also a group) that has any particular advantage over a true group.

Now, what's a more interesting (and considerably harder) question is "what properties do you need for DHKE to be secure?"  We do have assumptions such as the CDH assumption ("given $g, g^a, g^b$, it's hard to compute $g^{ab}$), however we don't know what semigroup properties ensure that...