I'm confused about the ontological status of a group, for lack of a better word. My confusion is best illustrated with an example:
Suppose I give you an "abstract group" $\mathbf{G}$ with the exact same multiplication table as the set $\{\pm 1 , \pm i, \pm j, \pm k\}$ (quaternions) under multiplication. And someone says find a homomorphism for this abstract group.
Since I know it follows the same multiplication table as the set I gave above can I suggest the appropriate bijection for those two groups, and say I've constructed the homomorphism?
What doesn't convince me is that basically I'm saying "the so-called quaternion group is isomorphic to the set of base quaternions under multiplication" which sounds pretty trivial.
So finally, my question in its most succinct form boils down to:
Is it correct to suggest a homomorphism of an abstract group to one of its realizations, or is this not done since its pretty trivial? Are homomorphisms only constructed between abstract groups?
To give another example, I give you a $D_4$ table and you suggest to me that an isomorphism is the group of rotations and reflexions for a square. Isn't this a triviality?
Is a group something "above" concrete examples and as such it is only proper to compare abstract groups with other abstract groups?
Note: I might be suffering from the fact that I haven't seen enough examples and only been exposed to abstract algebra in an abstract way.
Thanks.
A group is well defined in mathematics and there is no significant distinction between an "abstract" group and a "concrete" group. A group is an ordered pair $(G,\cdot)$ where $G$ is a set and $\cdot$ is a binary operation $G\times G\to G$ satisfying the appropriate axioms.