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.

  • Some group isomorphisms are trivial like $2x = x + x$. Others are less trivial; maybe like $e^{i\pi} = -1$. – pjs36 Oct 13 '15 at 23:45
up vote 2 down vote accepted

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.

  • So can Q8 be isomorphic to the set of quaternions I posted above even if saying such a thing sounds trivial? – DLV Oct 13 '15 at 23:38
  • 2
    @SofiaV For groups to be isomorphic it means there is a bijective function between them preserving the operation. I could make a quaternion group out of hats and chickens, as long as I know how to compose them with the binary operation. Nothing has to physically happen; the operation is just a rule for taking two things and getting one thing from them. – Matt Samuel Oct 13 '15 at 23:40
  • I think you're answer can be contradictory with mathworld.wolfram.com/AbstractGroup.html – DLV Oct 15 '15 at 0:03
  • Do you have any reference for "the opposite of abstract group is permutation group"? I can't find it. Thanks. – DLV Oct 15 '15 at 0:06
  • Also, en.wikipedia.org/wiki/Group_theory#Abstract_groups, does indeed form a distinction between abstract and concrete groups. – DLV Oct 15 '15 at 0:12

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