Welcome back, STEEMathematicians! This post marks the second installment of our introductory series on the truly elegant field of group theory. Last time, we introduced the basic notion of a group, together with the core, defining group axioms. We then considered some basic examples of groups, before boldly tackling and understanding the logic behind some key elementary results. If you were unlucky enough to miss out on the first post, you can find it here. In today's post (as promised!), we will begin by considering an illuminating example of a permutation group, before moving on to think about mappings between groups. So, shall we?
The Rubik's Cube group is a group that represents the structure of the well known (and loved) mechanical puzzle. Each element of the set corresponds to a cube move which can be any sequence of rotations of the cube's faces.
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Permutation groups have the interesting, and tremendously useful characteristic property that facilitates the reordering of a finite number of objects; the exact number of which is called their degree. This reordering can be expressed succinctly as cycles, denoting the sequence of the replacement of elements. The notation (abcd) represents a cyclic operation in which a is replaced with b, b with c, c with d, and d with a, closing the cycle.
Now, it is the case that for a permutation to be defined, the operation on all elements must be defined. We can say, therefore, that a permutation is a combination of cycles, denoted by, so-called attached cycles, such as (abcd)(e)(f) for a 6 object permutation.
Example: (142)(35)(6) reorders {1, 2, 3, 4, 5, 6} to {4, 1, 5, 2, 3, 6}.
The order of a permutation is easy to determine from the cyclic notation - it is the lowest common multiple (LCM) of the cycle lengths; so, in the afformentioned example, the order is 6, as we would expect. Ace.
This is a particularly neat concept - as we go through the following definitions, try to imagine (vividly, no doubt!) what is going on! For me, groups are a hazy, green, poorly-defined shape, and I imagine mappings as scattering the group elements beyond the bounds of its shape, forming some other structure. Think of your own way to visualise it - it's fun!
Now, lets be a little more serious by defining a mapping Φ, between two groups G and G'
Φ : G → G'
maps every element X of G onto an element X' of G', written as X' = Φ(X). We refer to X' as the image of X under Φ.
G and G' are homomorphic if there exists a mapping Φ, such that
Φ(X) · Φ(Y) = Φ(X · Y)
Φ is then called a homomorphism. Homomorphisms satisfy the following properties:
Identities are mapped onto eachother: Φ(I) = I', as X' = (IX)' = I'X', where we have simplified our notational convention via Φ() = ()', and omitted the product sign.
Maps inverses: I' = (XX-1)' = X'(X-1)', thus (X-1)' = X'-1.
If X has the order m, then X' also has the order m, or a factor of m: I' = (Xm)' = (XXm-1)' = ... = X'm.
G and G' are isomorphic if there exists a homomorphic mapping Φ with a one to one correspondence of elements, i.e. Φ(X) = Φ(y) implies that X = Y, and Φ(G) = G'. In this case, we refer to Φ as an isomorphism.
Isomorphic groups have the same number of elements, and also the same number of elements of given order m.
For a homomorphism Φ, the set of all elements that map to the identity element ({X}, for which X' = I') are called the kernel of Φ. Isomorphisms have only I as kernel.
We consider the real numbers ℜ under addition, and the complex numbers with unit modulus U(1) under multiplication. The defined Φ is a homomorphism:
(x + y)' = ei(x + y) = x'y'
We see that the elements x + 2πn, with n ∈ ℤ, are mapped to x'. This is therefore not an isomorphism, and the kernel of Φ is {2πn}.
A homomorphism Φ for which Φ(X) = Φ(Y) implies X = Y is called a monomorphism.
A homomorphism Φ for which Φ(G) = G' is called an epimorphism.
If G and G' are the same, Φ is called an automorphism.
Right, that's all for today folks - short and sweet! There's a lot to think about there, so I suggest that you work through this post with a pen and paper, if this is your first time meeting the topic, as the concepts are relatively subtle. If you're feeling adventerous, see where your intuition takes you, using what we have covered so far and see if you can find any interesting results! In the next post, we will start thinking about subgroups, and we will use this knowledge in figuring out how to divide a group using equivalence relations. As always, if anyone has a question, please do feel free to ask away in the comments section, and I will do my best to get back to you. Until next time.
