In the last post, I talked about what symmetry means. A symmetry is an immunity to some kind of transformation. But I left the idea of transformation informal and intuitive. In this post, I’m going
to move towards formalizing it.
The group theoretic notion of immunity to transformation is defined in terms of group isomorphisms. A group isomorphism is a structure preserving mapping between two different groups. If you know category theory, it’s defined very easily: a group isomorphism is an iso arrow in the category of groups. Of course, that’s a bit of a hand-wave, because I haven’t explained just how to define the category of groups. To do that, I would need to explain how to structure the group, and what all of the arrows are.
The category of groups, commonly called Grp, contains all groups as objects. There are arrows between groups if and only if there is a homomorphism between the corresponding groups. Given two groups,
(A,+) and (B,×), a function f : A→B is a homomorphism if and only if ∀ x,y ∈A : f(x+y) = f(x)×f(y).
What that means, informally, is that the mapping from the group A to the group B preserves the
group-theoretic properties of A: I can apply the group operation of A on any two elements before mapping to B, or I can apply the mapping to B and then use B’s group operation, and I’ll get the same result. So the group-properties of A are embedded in B. The fact of the structure preserving properties
of the homomorphism means that f preserves identity – that is, f(1A) = 1B, and it also means that the mapping preserves inverses: ∀x∈A, f(x-1) = f(x)-1.
On the other hand, there are some properties that aren’t preserved by a homomorphism. A homomorphism
isn’t onto: while every member of A is mapped to a member of B, not every member of B is necessarily
mapped onto by a member of A – so there are members of B that have no corresponding value in A, and B’s
group operation doesn’t have to preserve A’s structure when you perform an operation using one of those
values. The mapping also doesn’t have to be one-to-one: multiple members of A can be mapped onto a single
element of B. Any distinctions between those values is (obviously) lost in the mapping.
If we fix those two weaknesses, by requiring that the mapping be one-to-one and onto, then f is an
isomorphism from A to B. If there’s an isomorphism from A to B, that means that there is a fully
structure-preserving mapping between A and B: A and B are equivalent. You cannot tell the difference
between the two of them using their group operations.
To drop back to the category theory for a moment: saying that to be a isomorphism, a homomorphism
must be one-to-one and onto is really just another way of saying that in the category of groups, Grp, where homomorphisms are arrows, an isomorphism is an iso-arrow. An arrow is iso in category theory when it’s got a particular kind of relationship with identity; if there’s an isomorphism between A and B, then A and B are cancelable by an arrow in the category: if there’s an iso arrow from
A to B, and there are homomorphisms from G to A and B to H, then
there’s a homomorphism from G to B and from A to H; the step from A to B can be canceled by a composition that reduces it to identity.
An easy way to understand this is to look at one of the most canonical examples of group theory: permutation groups. Permutation groups go right to the heart of group theory; they’re the first groups
that were studied; in fact, you could reasonably make an argument that group theory was originally developed specifically to study permutation groups.
A permutation group is a group where you can re-arrange the elements without creating
a visible difference. Suppose you’ve got a pentagram: that is, a graph made of five points, where every pair of points is connected by an edge, like the diagram to the right. This can form a group P, where the edges of the pentagram are the elements of a group. The group operation of P is edge-sums: if there’s an edge ab and an edge bc, then ab+bc=ac.
An isomorphism from a pentagram group to itself is a permutation: that is, a mapping that re-arranges the edges. You can map any edge of the original group to any other edge. As long as the mapping is total, 1:1, and onto, after you’ve finished the mapping, you’ve still got a pentagram group. You can’t tell the difference. You can, for example, switch A and C in the pentagram, which will reshuffle a bunch of edges – but when you’re done, there’s no way to detect that you’ve changed anything. It’s still a pentagram; there are still edges ab, ac, ad, and ae; and those edges still get the same answers in edge-sums. It’s indistinguishable from the original.
Getting back to the original point, we can now say exactly what we mean by symmetry in group theory. A transformation of a group is a group isomorphism: an operation that changes the group, mapping it onto either another group, or a permutation of itself; and after that mapping, the result is indistinguishable
from the original group.
There is, of course, a bit more to it. (Isn’t there always?) I’ve said that the simple addition group
can be considered a definition of mirror symmetry; but with what I’ve explained so far, there’s no way of using the addition group to describe mirror transformations of anything but real numbers. Clearly a proper definition of mirror symmetry needs to be able to work on more than integers; and we’d certainly like to
be able to have a single definition of it that works not just on numbers, but on anything where
we can observe a kind of mirror symmetry. That’s the topic of the next post: group operations.