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Many commutative groups could be imagined as something "concrete", for example $\mathbb N$ as an abstraction of operations on sets of objects, and from this $\mathbb Z, \mathbb Q$ and $\mathbb R$ are constructed; also if I have some objects and rules to combine them, something like an algebraic structure arises. This is one thing to see it.

Another is more abstract, and views algebraic objects as permitted actions on other objects, in the group setting I mean permutation groups (as this view corresponds to mappings, I think the associativity is crucial here; and given that, in essence both views are equivalent in that a mapping, or set thereof, could be regarded as an object of its own).

Now when I think about non-commutative groups, like $\mathcal S_3$ and many others, just their realisation as permutation groups, i.e. symmetrie groups of other objects, or by explicitly listing them by generators and relations came to my mind.

But are their any concrete, real world examples where non-commuting finite groups are realised as objects on their own (i.e. without seeing them as operations on another object)? What I mean is where in some sense the objects are "experienceable" in the sense that they exist materially, and there are certain operations to combine them?

I hope my question is not to vague and it is clear what I am asking about?!

Bernard
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StefanH
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  • Any finite group is isomorphic to a group of (real) square matricies. Reading your post, I honestly have no idea whether this is going to count as "concrete" or not... – David C. Ullrich Jun 10 '16 at 18:25
  • I don't really understand the question. If you aren't realizing them as symmetry groups, how are you "experiencing" the group operation? – Qiaochu Yuan Jun 10 '16 at 18:30
  • @QiaochuYuan meant the group elements to exists as objects (or equivalence classes of objects), for example take a paint box, then we have colors and we combine them by mixing them, giving an operation. Computing with numbers could be realised as performing operations with sets (conceptually, which represent numbers). But having something exists just as permittable operations on another set is something more abstract, but I do not want this extra layer of abstractness, I am looking for concrete examples, preferable exist materially, which you can feel with your senses from first principles. – StefanH Jun 10 '16 at 18:37
  • @DavidC.Ullrich Mhh, to abstract, as matrices do not exists in nature, they just exist in our heads to organize data or represent mappings, and in some sense representing group as linear mappings is like representing them as subgroups of symmetric groups. – StefanH Jun 10 '16 at 18:38
  • @Stefan are the integers/reals really fundamentally different from matrices though? I have never seen the 'object 1'. I have at times seen a single apple, but that is not '1'. So I'm not sure I agree that the integer 'do exist in nature' – user2520938 Jun 10 '16 at 18:54
  • @user2520938 They are direct abstractions of concrete objects. Of course they are not concrete in itself, but numbers are an equivalence class of sets (or a representation of this class) with respect to their cardinality; in such a single apple is one representant of $1$, as is a single house, or any singleton set. So I would regard them as experiencable. – StefanH Jun 10 '16 at 18:57
  • So how are the rationals and reals experienceable? (And what is -2 houses?) Given that this is your prime example of what you want it would be nice if you were completely clear. It seems like you are asking for different things, representing groups as completely abstracted objects, but have them tangible. I don't know how any of your example groups exist materially. –  Jun 11 '16 at 14:21
  • Okay, the natural numbers are experiencable, and in some sense rationals are as proportions; I admit negative numbers and the reals are not, but they are more or less directly derived from the naturals and the rationals, in a more direct way than groups which act on some object. – StefanH Jun 12 '16 at 16:32

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I think that finite symmetry groups do qualify, e.g., the dihedral groups $D_n$. Polygons are "real", although they might not "exist materially". However, groups are rarely concrete without "another object". I think of symmetry groups of codes, for example. The Mathieu groups arising by the Golay codes. This is great ! Why should I expect that the Mathieu groups appear more concrete in nature ? Rubin's cube does exist materially, and hence a lot of finite non-abelian groups with it - see this question.

References: For a similar question, see this MO question.

Community
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Dietrich Burde
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    I think the point of the OP is that while polygon are 'concrete', rotations/reflections of polygons are not; they aren't 'objects', they just act on other objects. In contrast to for example the integers, which in some sense have an existence independent of any other object; they are not inherently described as actors on some external object. – user2520938 Jun 10 '16 at 18:50
  • @user2520938 Yes, that's the point! – StefanH Jun 12 '16 at 16:32