We have looked at symmetries of objects such as polygons in our previous postings; however, we haven’t looked at what it would mean for a set of elements to be symmetric. That is, if I have a collection of objects other than shapes in the plane, what would it mean to say that the collection is symmetric?
As an example let’s look at the English alphabet, is this collection of letters symmetric? To begin with, let’s ask a simpler question. If I rewrote the alphabet as b,a,c,d,e,f,…,z instead of a,b,c,d,e,f,…,z, that is I switch the order of a and b but leave everything else the same, would it change the alphabet or the totality of the English language?
If we only consider the letters as names, I would claim that it wouldn’t change anything. While we may have to write ‘abck’ instead of ‘back,’ if we decided that the b fulfilled all usual roles that a does including sound, then the word ‘abck’ would sound like what we call the work ‘back.’ Therefore, if we were to relabel the alphabet interchanging a and b, we would still arrive at a set that would meet our needs of an alphabet. In this way, I would say that the alphabet has a symmetry created by the interchanging of a and b.
In the previous example, there is nothing special about a and b. Therefore, we can interchange any two letters and have a symmetry of the alphabet. Generalizing this, if we take any mapping, f, from the alphabet back to the alphabet, where we permute the letters of the alphabet, we will indeed have a symmetry of the alphabet since the resulting naming of letters would fill all the needs the original alphabet had. Since, any element of the alphabet can be mapped to any other under a symmetry, we would say that the alphabet has high symmetry.
In order to further justify my claim that the renaming of letters wouldn’t change the meaning of words, I refer to Shakespeare that so eloquently wrote
A rose by any other name would smell as sweet.
Now suppose that we look at the set of all English words. Since every word is created using letters written in order, we could obtain the set of all possible English words by starting with the alphabet and defining an operation of concatenation of letters on the resulting set. That is, if we take a*b we get simply ab. Furthermore, if we take (a*b)*c we get ab*c which is abc. I note that, in this way, we define all possible words, not just words that are defined. If we now focus on the set of words, what symmetries of this set be?
Well, we would still want a that any symmetry would map one word to another, and no two words should go to the same word. I would further suggest that if we mapped ‘a’ to ‘i’ and ‘n’ to ‘t’, then we would also want our symmetry to map ‘an’ to ‘it.’ That is, we would want our symmetry to respect our operation of concatenation. Because of this, any symmetry must send each letter somewhere, so any word must be sent to the word created by changing the letters according to the rule for the letters. Also, since the symmetry must map something to everything, we would have to have that each letter would be mapped to another letter, and from the description of what the symmetry does to the letters, we can determine what it does to the words.
If we combine the first example with this new observation, we could find a symmetry which sends ‘the’ to ‘and’ by choosing a mapping that sends ‘t’ to ‘a’, ‘h’ to ‘n’ and ‘e’ to ‘d’. Furthermore, we would have to have these mappings for t,h and e, so it must be the case that ‘thee’ would be mapped to ‘andd.’ What we have found is that the set of symmetries of all possible English words is exactly the same as the set of symmetries of the English alphabet.
In general, we have decided that a symmetry of a set is a mapping which sends everything in the set to one thing in the set, exactly one thing in the set is mapped to each thing in the set, and the mapping preserves any properties (operations) we have defined. Such a mapping from a set to itself is called an automorphism. Therefore, we have a symmetry of a set whenever we have an automorphism from a set to itself. Furthermore, if a set of elements is generated by a set, then the set of automorphisms can be completely determined by the automorphisms of the generators.
Now, using these rules, can you find the set of symmetries of the natural numbers (1,2,3,4,…) with the operation of usual addition? What are all symmetries of the integers (…,-3,-2,-1,0,1,2,3,…) under the operation of addition? What if we defined the operation of take the minimum of two numbers on the integers, what would the resulting set of symmetries be?
I’ll leave these as open questions, so please let me know below if you find them. Bonus points if you can prove your assertion. I do want to note, however, that the only symmetry of one of these examples is the identity mapping. When this is the case, we would say the set is anti-symmetric. If you enjoyed the post, let me know by liking it and follow the blog to get updates on when more symmetry posts are made.