Group Theory 5: Subgroups, Conjugacy and Normality
14 June 2008 – 12:28 pm by Rich CochraneOne of the things mathematicians soon learn to look for when they meet a new mathematical object is its “subobjects”, which are parts of the object that have the same type of structure as the whole thing. In our case those are “subgroups”, which are subsets of a group that are, themselves, groups.
A Definition and an Example
We’ve already met an example of a subgroup. When we looked at the symmetries of the square, we first looked at the rotations, including the identity element, and then introduced “flips” (mirror reflections) as other symmetries. The rotations and the flips (and the identity element) taken together form the dihedral group D4. The rotations (and the identity element) alone are obviously a subset of this group. We can see that they’re a subgroup by checking three things:
- Closure: The composition of a rotation with a rotation is another rotation (never a reflection). To see why, imagine a square that was black on one side and white on the other, and imagine it’s lying black-side-up on a table. Rotations always leave it black-side-up, but a flip leaves it white-side-up. No amount of rotating can flip it over.
- Inverses: For every rotation, there’s another rotation that’s its inverse. No rotation can be undone by flipping, and again you can picture the black-and-white square to see why.
- Identity: We’ve said the identity element is in our set by definition. You need that, of course, to create a group.
These are three of the four group axioms; we don’t need to check associativity, fortunately: we know the operation is associative because D4 is a group, and the operation is inherited from that. We’ll see lots more examples of subgroups in this mini-series, but for the remainder of this instalment we’ll be constructing some theoretical machinery that we’ll find very useful later.
Conjugacy
It’s about here that group theory tends to lose people, so hang onto your hat. We’ll work with a general group called G for now.
Let g be an element of G. Now, for every element h of G, there’s another element of G,
h-1gh
called “the conjugate of g by h”. If g and h are operations like symmetries you can read this as “do h, then do g, then undo h again”. This is a fundamental idea in the solving of Rubik’s Cube, where many moves are constructed using conjugate pairs of simpler moves.
We actually use conjugation all the time in solving three-dimensional problems. If I want to put a “return to sender” address on the bottom of a parcel I’m wrapping I’ll turn it over, write the address, then turn it back over again. I’m not left in exactly the same position as before, because now my address is written on the bottom. Take something square and try some conjugations out, especially conjugating rotations by flips, and vice versa. Watch carefully where one of the corners ends up to see what’s happening.
Conjugation captures something that has intrinsic meaning, but we’ll be using it to define something else that has much more theoretical importance.
Note on Multiplicative Notation
From now on our work will be cleaner if we drop the special symbol for the group operation and put two letters next to each other with the operation being understood:
aˆb = ab
In our new notation, we already know (by definition) that if e is the identity element then, for any group element a,
ae = ea = a
and
aa-1 = a-1a = e
But in general for group elements a and b we do NOT have
ab = ba !
unless the group is known to be Abelian. If we do have an equation that we know, such as
ab = c
then we can “multiply” both sides by the same group element and we get another equation that’s true:
abd = cd
or, more enlighteningly,
abb-1 = cb-1
which implies that
a = cb-1
We’ll see these kinds of maniplulations a lot in the rest of this series, including the next next section, where we’ll use them prove something fairly important.
Conjugacy Classes
Take any two elements of the group,, say g and h again. It might be that there’s an element, say x, that conjugates g to h:
x-1gx = h
Then again, there might be another element, say y, that conjugates h to g:
y-1hy = g
What we’d like to know is whether one implies the other — that is, whether “g is conjugate to h if and only if h is conjugate to g”. If so, we can just talk about g and h having a special relationship of being conjugates of one another.
The proof that this is in fact the case is a nice example of how elementary proofs in group theory are done. We start with the assumption that
x-1gx = h
and we aim to get g on its own. Well, we know that
xx-1 = e
where e is the identity element, so we’ll use that to simplify this
xx-1gx = xh
to this:
gx = xh
Now I’m sure you can see the next move — apply x-1 to both sides on the right:
gxx-1 = xhx-1
and again cancel the xx-1:
g = xhx-1
There — not only have we worked out that the necessary element y exists, but also that y=x-1.
This means that “being-a-conjugate-of” is a reflexive relation, like “being-a-sibling-of” but unlike “being-a-parent-of”. If you’re my sibling, I must be your sibling too. Likewise, if g is conjugate to h then h must be conjugate to g.
If you know what an equivalence class is then you’ll know where we’re going next. What we’d like to know is whether, like the sibling relationship, conjugacy is “transitive”. Say we know that g is conjugate to h and that h is conjugate to j. Does that mean g is conjugate to j?
Let’s again do this by calculation, but I’ll leave you to see how we arrive at each step.
x-1gx = h x-1gx = y-1iy yx-1gxy-1 = i
By closure, yx-1 is an element of the group — call it z. It’s easy to see by calculation (try it) that xy-1 is z-1; hence g is indeed conjugate to i.
By analogy, we know that if Bill is Bob’s brother and Bob is Bert’s brother then Bill is Bert’s brother too. This enables us to just put them all into a class called, say, the Jones Brothers. In our case, this is called a conjugacy class.
It’s a fact of basic set theory that a relationship like this — which is called an equivalence relation — “partitions” the set on which it’s defined into subsets so that every element of the main set is in exactly one subset. So it is for us: every group can be partitioned into conjugacy classes, so that every element of the group is in exactly one conjugacy class, the one containing its group-theoretic brothers and sisters.
Normal Subgroups
There’s a certain kind of subgroup that will be important to us soon, and although in a way it’s a “special” kind of subgroup it’s called, with typical mathematical perversity, a “normal” subgroup. Normality can be defined easily enough:
A subgroup H of a group G is normal if H is a union of conjugacy classes of G.
Now what’s happened here is that we’ve got a bit abstract. Probably you’ve got a handle on conjugates, and you might feel like you can picture conjugacy classes as dividing a group up into “categories” of elements. But this definition probably seems arbitrary and, well, a bit weird.
Let’s look at what it means. First, any subgroup of a group must, in order to be a group at all, contain the identity element. And it turns out that the identity element is always in a conjugacy class all on its own. Imagine it wasn’t, and there was some element g that was conjugate to e but not equal to it. Then:
x-1gx = e x-1g = ex-1 x-1g = x-1 g = xx-1 g = e
which contradicts our original claim that g isn’t equal to e. So the conjugacy class containing e is just {e} (you can prove for yourself that, at least, every element is conjugate to itself).
Think of the symmetry group of the equilateral triangle, the dihedral group D3. This consists of the identity element e, two rotations r1 r2 and three reflections (”flips”) f1 f2 and f3, as you can check for yourself. Its conjugacy classes — and again you can check this — are
{e}
{r1, r2}
{f1 f2 and f3}
Looking at all the possible unions of these, we see that
{e} U {r1, r2}
is a group (again, you can check) and so by definition it’s a normal subgroup. On the other hand
{e} U {f1 f2 and f3}
isn’t a group because it’s not closed. Finally,
{e} U {r1, r2}
U {f1 f2 and f3}
is a group, so every group is a normal subgroup of itself. That sounds weird, but it neatens things up for us in some cases.
The point of defining normal subgroups isn’t just to annoy you with some new terminology. They have important connections with quotient groups and homomorphisms, which we’ll study very soon. In the next instalment we’ll define cosets and prove Lagrange’s Theorem, which is about subgroups and gives us another view of the normality criterion.
The image of the neatly-sorted Lego equivalence classes is courtesy of repoort.