I believe that magma isomorphism is defined as $\phi(x*y)=\phi(x)*'\phi(y)$. The automorphism group is the set of bijective isomorphisms from the elements of the magma to itself, under the operation of function composition. I take a magma to be just a set $X$ together with any operation from $X^2$ to $X$.
The reason I suspect that this might be the case is that:
- I know that groups are closely related to permutations, which is exactly what automorphisms are. If a permutation leaves something invariant, then it is not hard to generally show that its inverse does so too. The axioms of closure and inverse are met, and the requirement of identity is obviously met. Function composition is always associative, and so these functions do indeed form a group.
- These groups seem to potentially have the quite complex structure, and so few visible constraints that it just seems reasonable that any group could be like the automorphism group of a magma, but I don't believe that I am experienced enough in the ways of abstract algebra yet to prove this.
I'm just looking for a better way for myself to think of groups in general, that's all.