Just like Topologists can’t distinguish between a coffee cup and a doughnut(because they are homeomorphic), Group theorists have a hard time distinguishing between groups that are Isomorphic!
we will see example of two (or more) Groups, beside coming from different origin they are same in respect of Group theory.
I am omitting the basic definition of Group because I think there quite a few very good introductory content already exists, so without further ado let’s start.
the first Group . . .
this first Group comes a lot in Number Theory and its fairly easy to understand. just like integer can be categorized in to even and odd based on the reminder when divided by 2, we can categorized every integer based on reminder when divided by 3, but unlike even and odd we will name them [0], [1], [2].
as examples { -3, 0, 3, 9, 18, 51} are [0] type number because they can be divided by 3 , also {-4, 1, 10, 19} are [1] type number because they have 1 as reminder when divided by 3 and { -5, 2, 11, 20, 52 } are [2] type numbers because they have reminder of 2.
now if you add [1] number with a [2] number you always get a [0] number, you can check it out for yourself. in fact they have their own little arithmetic world in addition which can be seen in a neat little table obviously call Cayley table, so this set { [0], [1], [2] } with respect to + called Z3 or Zmod3. You can verify if it’s a group or not from the axioms of groups, its a fun little exercise.
Groups are mini model of algebra that’s why they with their bigger cousins like Rings and Fields are called Abstract Algebra
the second Group ….
what are the solution of this equation f(z) = z2 – 1 = 0 ? you might immediately tell its just 1, but what if the inputs are not only real numbers but complex number .this are numbers in the form of a + bi where a and b are real number and i2 = -1. the input domain the whole 2d plane instead of one-dimensional line.
we won’t be talking about complex function in this post. our original question if we take input complex number can we find other solutions of this equation z2 – 1 = 0 beside 1 ? yes there are two other solution and they are −½ + √(3/ 2)i and −½ – √(3/ 2)i . for convenience we will call them ζ1 and ζ2 respectively.
now we have three roots or solution of our equation ,lets name them C = { 1, ζ1 , ζ2 }. now notice that if we multiply ζ1 * ζ2 we get 1 back ! you can tell where I am going this, the set C with multiplication ” * ” form a Group
Isomorphism is…
Now I am putting two previous Group table side by side. Can you see that resemblance? they look exactly similar! yes, the have different operations, also they are different object altogether but how are they arrange in the table is exactly same!
now we can make a fairly easy function that takes each element from Z3 to C,
f : Z3 → C
we can literally draw out this function. And this f is the Isomorphism between Z3 and C. If you know a little bit of Group Theory you can see
the identity element mapped into corresponding identity element, the inverses also mapped in corresponding inverses, overall f preserves the essential structure aka the identical Cayley table.
That’s why a Group theorist is as just confuse as a Topologist when two Groups are Isomorphic
in last post about Homeomorphism I talked about Continuous function a lot, which in a sense is less restrictive version of Homeomorphism, Group Theory also has kind of analog of it which is called ‘Homomorphism‘ (don’t get confused , it doesn’t have that e in it ! )
Now that we have a little idea what Isomorphism is we can taste Cayley’s theorem. this is a neat little gem of Group Theorist .but we still need to know about one more special race of Groups called Symmetric Group (which is just basic combinatorics)
Cayley said , “God, give me every Group, I want to see them all”. God only gave Permutation Groups. Cayley smiled and replied ” Enough”
Suppose we have three different shape. how many ways we can permute them? its a classic combinatorics question and the answer is 3! = 6.
now we want to make a Group out of this 6 elements. We want a procedure to combine any two permutation to produce another permutation. one way is take a standard permutation as identity and use that as reference. lets take (♦ ♥ ♣) as identity, now combining two permutation is to change the left permutation as if it was the identity according what the right permutation is telling.
the point is its a valid group with its own operation and Cayley table. In general you can take n elements and n! permutations forms the Symmetric Groups S3 we will see that this are very universal type of Groups.
Cayley’s Theorem
Now see this neat little trick ,take the Cayley table of Z3 and separate out all the rows, observe that these are just permutation of these three elements, also that these permutation we all ready know they are the subset of our Symmetric group S3
So our two groups are nothing but a subgroup of a Symmetric group!
they are Isomorphic !
You can take any group of n element take the Cayley table of it, separate out the rows , they will be some permutations among the n! permutations, a subgroup of S3
so any group is isomorphic to a subgroup of Sn! you can construct yourself by taking its Cayley table . And its the proof of Cayley theorem!
Cayley’s Theorem : any finite group G is Isomorphic to a subgroup of S3.