I’m not feeling social interaction today so I learned a bunch of group theory, and it just got too cool, so i had to post some of it on here!
Now, before I go on to talk about the main things which are Isomorphisms and Permutations, I should properly define groups.
Taking the axiomatic approach, groups are sets with an operation for combining elements which pass the following tests
Groups are closed under one operation, meaning if you combine any two elements using an operation of your choice (the symbols +and * are often used), you will never be able to create an element which is not in the group.
The operation is associative meaning A*(B*C)=(A*B)*C=A*B*C as long as you combine the elements as shown. This effect can be extended to any string of n elements.
There exists an element “e” such that x*e=x and e*x=x. This element is called the unit, and if we are talking about multiplication with ordinary numbers, this number would correspond to one.
For each number in a group, there will be an inverse. When we are using multiplication-like operations like we are above, we usually write the number like this a-1, and call it “a inverse” for addition we use -a and call it “the negative of a”. The inverse of the unit is the unit again, and any element combined with its inverse becomes the unit.
You can see that the commutative axiom is not listed, where A*B=B*A this is not always true, as we will see in a second. Take the group consisting of the elements
under the operation of matrix multiplication which works like this.
Now, if we do the operation A*B, we get the matrix C, if we do the operation B*A, we get the matrix K, meaning that this operation is not commutative. This is not the only operation which behaves this way, so in general we have to say that the operation does not have to commute. If it does, we call the group an Abelian group.
Now that that’s out of the way, we can define an Isomorphism.
An Isomorphism is a one-to-one function which takes one group and converts it into another while preserving the operation. Basically, if G is a group under the operation * denoted <G,*> and <H,+> is a group, and
then an Isomorphism f(x) will take f(a)=a’ and f(b)=b’ and make, f(a*b)=a’+b’ or, entirely in function notation, f(a*b)=f(a)+f(b).
To prove that two groups are isomorphic, you need to prove that the function you are using is one-to-one, that is proving that the function is both injective and surjective. To do this, you have to prove that each element of the range is the image of at most 1 element of the domain, which can be done by showing that it fits this definition
You then have to prove that the function is surjective by showing that for some element y in he range, there exists an element who’s image is y. Together, these two things prove that the function is bijective, which means that you can turn elements from G into elements in H and vice versa.
From there, all you have to prove is that f(a*b)=f(a)+f(b) and you’re done.
Now why are isomorphisms important? Because they can be used to prove things like this Cayley’s theorem which says that every group is isomorphic to a group of permutations. Quite a bit is known about permutations, which makes this theorem very strong in terms of its use.
We can prove this in a second, but first, we have to look at permutations.
A permutation is a function on a set S which takes the elements of S and rearranges them. Now, this must by definition be a bijective function, which takes S and maps it to its self. Taking individual permutations to be elements of a set, we can create a set of n! elements, where n is the number of numbers being used in the permutation. For instance, the set of all permutations on three numbers is:
and has 3!=6 elements. To combine the elements, we define the permutation of a permutation to be a function which takes the elements as they are arranged by the inner permutation and then rearranges them under the outer permutation called the composition. For instance:
To show that this combining of permutations is closed under the operation above, we can make a chart like the one below, where the first column is multiplied on the left by the first row,
but there are an infinite number of groups of permutations, so we must prove it. We can actually take it a step further into abstraction, and prove the composite of any two functions which are both bijective will yield another bijective function. Why are we proving the combinations of permutations are bijective? Because we started with the assumption that we had made a list of all the possible permutations for n objects, so showing that this new function is also bijective permutation means that it was on this list in the first place.
Assume a f(x) and g(x) are injective functions, then we have to prove that [f*g][x]=[f*g][y] implies that x=y. Well suppose [f*g][x]=[f*g][y], then f(g(x))=f(g(y)) because f(x) is injective, g(x)=g(y), and because g(x) is injective, x=y, so [f*g][x] is injective
Assume g:A ->B and f:B ->C are surjective, then we have to prove that every element of C is f*g of some element in a. Assume z is an element of C, then because f is surjective, f(y)=z for some y in B. Because g is surjective,f(g(x))=z so, for any element of C, there is an element for which z is its image under f*g, so [f*g][x] is surjective.
Now, assuming the functions f and g are both bijective, it follows from what was just proven that f*g is bijective as well.
By this, the group of permutations under composition is closed.
There exist inverses for each element, which return each permutation to the permutation
which also happens to be identity element. Finally, combining permutations is associative, so we have a group.
Now we can prove Cayley’s theorem which again states that each group is isomorphic to a group of permutations.
Begin by assuming we have a group G, we wish to prove that it is isomorphic to a set of permutations. Well, permutations require a set, so why not use the set that G is built off of? The only set we have around is the one used to make G, so we have no other choice but to use it. We then define a function
which is defined as:
And gives all the permutations of G, changing each element a of G into ax, and then ranging X over all elements of G to give one permutation per element.
This function is injective, that is
and surjective, as for some y in g,
so y is the image of some element a^(-1)*y.
So we have a bijective function which turns an element a of G into a permutation which turns x_1 into a*x_1, x_2 into a*x_2 and so on. So we let G* represent the set of permutations of G created by pi_a(x) as a ranges through the elements of G.
Notice now that the set of permutations created is not necessarily the set of all permutations of G, but a permutation corresponding to each element. We now have to prove that it is a group, that is
and that for any permutation contained by G*
Where pi_e is the identity permutation given by multiplying each x in G by the identity. The associative property can be assumed due to the fact that the operation is simple non-commutative multiplication.
To begin, we have to prove that
We can say that
Because ab is a member of G, we can say that G* is closed with respect to composition.
Finally, because G is a group with inverses,
So we have proved that G* is in fact a group.
Now all we have to do is prove that G is isomorphic to G*. We already have the function ready, so we use it as the isomorphism.
It is injective,
and surjective, every element pi_a of G* is the image of some f(a)
Thus, f is an isomorphism, and
That’s about all I have left in me for today, but I’ll try to get more up on here tomorrow! I thought Cayley’s theorem would be a good start!