In the last three videos, if we want to denote a sequence of symmetries done, we would draw
a little flow chart. But this is a little space-consuming, so we would abbreviate as
tau zeta sigma. It might be a bit strange that we read off the sequence of symmetries
from right to left. But when we think about how the symmetries act on x, it makes sense
to abbreviate this sequence this way. If we first let sigma act on x, we get sigma star
x, Then we let zeta act on this result, so it is now zeta star sigma star x, and finally
we have tau to act on this result. Now we have the symmetries in the exact same order
in the abbreviation.
Another abbreviation we would make is about undoing symmetries. Previously, we just say
undo sigma for a symmetry that undoes sigma, but we can abbreviate this to be sigma to
the power -1. We usually read this sigma inverse. Looking at these two notations, it seems like
we are multiplying symmetries, but please don’t treat it like normal multiplication.
To illustrate this, consider the symmetries of a regular hexagon, and denote r to be symmetry
that does anticlockwise rotation by 60 degrees s to be a reflection along the horizontal
axis. On the left we will do the symmetry s first then r, and on the right we will do
the symmetry r then s. We now place a red dot for tracking what these symmetries do.
On the left, we do a reflection first then a rotation, the red dot is up there. On the
right, we do the rotation first before reflection, and the red dot is down there. So these two
are different, so please don’t treat this abbreviation as just normal multiplication.
So we can talk about conjugation with more succinct notation. Conjugation looks something
like this. You put the symmetry h you want to conjugate in the middle, and you sandwich
h with g and g inverse, where g is the symmetry you conjugate h by. The reason why I emphasised
that this is not normal multiplication is that h after conjugation is not the same as
h itself. In fact ghg inverse and h are very similar to each other, though not necessarily
not equal to each other. But how similar are these two symmetries?
We will again use the example of the two symmetries of hexagon. And we consider the conjugation
of the horizontal reflection by an anticlockwise rotation. We first do a symmetry that undoes
r. This means a clockwise rotation of 60 degrees. Instead of actually doing a clockwise rotation,
we do something different this time. Imagine this is you, and you are facing to the left.
Now notice that the horizontal is the direction you are facing. So now to do this clockwise
rotation, we would shift our perspective instead. Rotating our perspective anticlockwise while
holding the hexagon still is the same as a clockwise rotation.
So now we need to do a horizontal reflection. Remember that the horizontal in your perspective
is the direction you are facing. So this green dashed line is the horizontal. So s here means
the horizontal reflection along this green line. So now, we need to do an anticlockwise
rotation, which is again the same as shifting the perspective. This time we shift the perspective
back to what we started off with. Now notice that all we have done is reflecting along
this green axis.
So before conjugation, s itself is a reflection along the horizontal axis, but after conjugation,
it becomes a reflection along the green axis. So s being conjugated is still s, just viewed
in a different perspective. And here, s is conjugated by r, and r determines how you
will change your perspective. So now we can answer the question of how similar these two
are. These are actually the same symmetries but viewed in different perspectives. I repeated
myself twice because this is a really important intuition to have for conjugation.
To make sure we get the intuition right, we revisit the hexagonal symmetries, but this
time we will conjugate r by s instead. According to our intuition, this is basically just another
anticlockwise rotation, in a horizontally reflected perspective. In a normal perspective,
this red dot will rotate anticlockwise with the hexagon. But now, in a reflection along
this horizontal axis, the rotation becomes clockwise. This means that this conjugation
results in the inverse of r, because it is a clockwise rotation rather than an anticlockwise
one. And this sort of relationship between these symmetries is now easily unveiled, while
previously you might need to see where these symmetries send each vertex to.
Conjugation gives rise to many other important concepts in group theory, and for the rest
of this video, we will focus on normal subgroups and simple groups, and leave the quotient
group concept for now, because I will cover it in the next video. But for the normal subgroup,
we first need to revisit the concept of a subgroup mentioned in just the previous video.
This was what we came up with when we discussed Lagrange’s theorem in the last video. To
make the concept of subgroup, and ultimately normal subgroups more concrete, here is a
non-trivial example. We will consider a subset of symmetries of cubes. Of course, the identity
has to be there. It is in one of the properties. Then another symmetry in this subgroup is
rotating by 90 degrees. We also have rotation by 180 degrees, as well as by 270 degrees.
To show that this set of smymmetries really is a subgroup, we first check that composing
any two symmetries in the set gives you some symmetries inside the set.
This is not difficult to check, because if we first do rotation by 90 degrees, then do
180 degree rotation, then this is the same as a 270 degree rotation. In fact you just
add up the angles, and subtract 360 degrees from it if necessary. And identity is visibly
in the set of symmetries here. So now it comes to undoing symmetries. Undoing identity is
of course still the identity, because we have done nothing in the identity symmetry. This
case is the same for 180 degree rotation. The way to undo a 180 degree rotation is to
do it again. And the remaining two undo each other: a 90 degree rotation undoes the 270
degree one and vice versa, and so this set of symmetries really is a subgroup.
Previously, when we think about conjugation, we only consider conjugating a symmetry by
another symmetry. But what about conjugating an entire subgroup by g? It simply means that
we conjugate each symmetry in the subgroup by the same symmetry g. The collection of
these conjugated symmetries will be denoted as gHg inverse. Like conjugation of a symmetry,
this resulting set of symmetries should be very similar to H itself. And in fact, this
set of symmetries form a subgroup itself, because these properties hold as long as you
view the symmetries in the same perspective. But these two subgroups are not always equal.
For example, when we look at the subgroup of symmetries of a cube we just discussed,
these symmetries are rotations about a vertical axis. But when you look at the symmetries
from above, these will become rotations about an axis pointing into or out of the screen.
These two subgroups are definitely different because the axes of rotation are already different.
So if we call this subgroup H, then this subgroup on the right is gHg inverse for some symmetry
g, and they are demonstrably different.
However, although in general they are not equal to each other, in some cases, they do
equal to each other. What I mean by them being equal is that for all little h inside the
subgroup H, conjugating it by g gives something in the subgroup H. We are not saying it has
to be h itself, just somewhere in the subgroup H will be fine. In this special case, the
symmetry g is said to be normalising H. And in the exceptional case where everything normalises
the subgroup H, or in other words, the subgroup H looks the same in every possible perspective,
we say H is a normal subgroup. There are always two normal subgroups, namely the set only
including the identity, as well as the whole group of symmetries itself. The reason for
this set to be normal is that the identity viewed in whatever perspective is the same
- doing nothing. And for the whole group being normal itself is because conjugating any symmetry
by any symmetry is still a symmetry, so it is still inside the group G. Lastly, if there
are no more normal subgroups of G, then we say G is simple.
It might be a bit tricky to remember the definitions of these new terms normal subgroups and simple
groups. But the simpler takeaway is that normal subgroups look the same under all possible
perspectives. And simple groups are simply those with a minimal number of normal subgroups.
The reason why we are so interested in these concepts is to allow the definition of quotient
groups and we will see in the next video, why this is a useful concept.
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