What does it really mean for a group to be "nilpotent"?
This year, I've asked many people to describe it in simple, memorable terms, and have yet to get a good answer.
Usually: something something about an ascending series. But what exactly, and WHY? Let's dig in! 🧵👇
1/17
This year, I've asked many people to describe it in simple, memorable terms, and have yet to get a good answer.
Usually: something something about an ascending series. But what exactly, and WHY? Let's dig in! 🧵👇
1/17
First, I'm wasn't at all picking on anyone, but rather, at how this concept (and so many others) are taught in nearly every algebra class and book.
By the end of this thread, you'll learn what nilpotent really means in a memorable visual way you'll never forget!
2/17
By the end of this thread, you'll learn what nilpotent really means in a memorable visual way you'll never forget!
2/17
First, a quick refresher. In the subgroup lattice of G:
--subgroups H≤G appear as down-sets, like stalagmites
--quotients G/N appear as up-sets, like stalactites.
Here are two groups of order 20. The dihedral group D_5 is a subgroup of one and a quotient of the other.
3/17
--subgroups H≤G appear as down-sets, like stalagmites
--quotients G/N appear as up-sets, like stalactites.
Here are two groups of order 20. The dihedral group D_5 is a subgroup of one and a quotient of the other.
3/17
Next, recall the commutator subgroup G'. It's the smallest subgroup such that G/G' is abelian.
In other words: "how far can you jump down the lattice so the resulting stalactite is abelian"?
In these examples, the quotients G/G' are:
C3 C4
V4 C4 x C2
4/17
In other words: "how far can you jump down the lattice so the resulting stalactite is abelian"?
In these examples, the quotients G/G' are:
C3 C4
V4 C4 x C2
4/17
I describe the operation of taking a commutator as a "maximum abelian step" down the lattice.
We can iterate this. The commutator of G' is G'', and so on.
A group is "solvable" if we eventually get to the bottom!
Let's do an example and a non-example.
5/17
We can iterate this. The commutator of G' is G'', and so on.
A group is "solvable" if we eventually get to the bottom!
Let's do an example and a non-example.
5/17
Example: the order-24 group SL(2,3) of 2x2 matrices over Z_3 is the smallest group that requires 3 "maximal abelian steps" to reach the bottom.
6/17
6/17
Non-example: The alternating group A5 is the smallest nonabelian simple group.
Since G' is normal, there are only two choices for it: {1} and A5. But the former is impossible because A5/{1}≅A5 is nonabelian.
Therefore, we get stuck:
A5 = G = G' = G'' = G''' = ....
7/17
Since G' is normal, there are only two choices for it: {1} and A5. But the former is impossible because A5/{1}≅A5 is nonabelian.
Therefore, we get stuck:
A5 = G = G' = G'' = G''' = ....
7/17
What does this have to do with nilpotent groups?
Solvability is defined algebraically. But that's not how you should think of it.
Solvable means: "if we climb DOWN the lattice, we reach the bottom."
Nilpotent means: "if we climb UP the lattice, we reach the top."
8/17
Solvability is defined algebraically. But that's not how you should think of it.
Solvable means: "if we climb DOWN the lattice, we reach the bottom."
Nilpotent means: "if we climb UP the lattice, we reach the top."
8/17
But just how DO we climb UP a subgroup lattice?
Start at the bottom, Z0:={1}. How do we find a canonical normal subgroup?
Every group has a center! Let's jump up to Z1:=Z(G) in the subgroup lattice!
Now, "chop off" the lattice below Z1, i.e., take the quotient, G/Z1.
9/17
Start at the bottom, Z0:={1}. How do we find a canonical normal subgroup?
Every group has a center! Let's jump up to Z1:=Z(G) in the subgroup lattice!
Now, "chop off" the lattice below Z1, i.e., take the quotient, G/Z1.
9/17
The quotient group has a center. By the isomorphism theorems (see previous images), it has the form G/Z2, for some subgroup Z2 of H.
Now, we have Z0 ⊴ Z1 ⊴ Z2.
Repeat this process: chop off the lattice below Z2 via a quotient G/Z2, and find the center of that: G/Z3.
10/17
Now, we have Z0 ⊴ Z1 ⊴ Z2.
Repeat this process: chop off the lattice below Z2 via a quotient G/Z2, and find the center of that: G/Z3.
10/17
In summary, to climb UP a subgroup lattice:
--start at the bottom, and jump up to the center.
--chop off everything BELOW to get a new lattice (a quotient!). Find the center of that group.
--repeat this process.
If we reach the top, then G is nilpotent!
11/17
--start at the bottom, and jump up to the center.
--chop off everything BELOW to get a new lattice (a quotient!). Find the center of that group.
--repeat this process.
If we reach the top, then G is nilpotent!
11/17
Theorem: nilpotent => solvable. In plain English, I remember this as:
"it's easier to fall DOWN than to climb UP"
Since p-groups have non-trivial centers, we immediately conclude that they're nilpotent, and hence solvable (which isn't a priori clear).
12/17
"it's easier to fall DOWN than to climb UP"
Since p-groups have non-trivial centers, we immediately conclude that they're nilpotent, and hence solvable (which isn't a priori clear).
12/17
There are many equivalent characterizations of what it means to be nilpotent, e.g.: "G is the direct product of its Sylow p-subgroups".
My favorite involves normalizers. If H isn't normal, then it's either "fully" or "moderately unnormal". Here's a picture:
13/17
My favorite involves normalizers. If H isn't normal, then it's either "fully" or "moderately unnormal". Here's a picture:
13/17
The index of the normalizer measures "how close to being normal" a subgroup is. This also counts the number of conjugate subgroups.
"Fully unnormal" means that the "fan" of conjugate subgroups is as big as possible. The only left coset that's a right cosets is H itself.
13/17
"Fully unnormal" means that the "fan" of conjugate subgroups is as big as possible. The only left coset that's a right cosets is H itself.
13/17
Theorem: A finite group G is nilpotent iff it has no fully unnormal subgroups.
In particular, p-groups cannot have fully unnormal subgroups. Their subgroup lattices group UP, rather than OUT.
14/17
In particular, p-groups cannot have fully unnormal subgroups. Their subgroup lattices group UP, rather than OUT.
14/17
In summary, a finite group G is:
SOLVABLE if we can climb DOWN its lattice via "maximal abelian steps"
NILPOTENT if we can climb UP its lattice via "jumping up to the center, chopping off below, repeat".
Theorem: "Falling down is easier than climbing up."
15/17
SOLVABLE if we can climb DOWN its lattice via "maximal abelian steps"
NILPOTENT if we can climb UP its lattice via "jumping up to the center, chopping off below, repeat".
Theorem: "Falling down is easier than climbing up."
15/17
When I showed this to a PhD student last month, I got one of the most audible "AH HA!"s I can ever remember.
Moral: concepts in algebra, and so many other areas of math, NEED to be motivated visually. Unfortunately, this is the exception rather than the norm.
16/17
Moral: concepts in algebra, and so many other areas of math, NEED to be motivated visually. Unfortunately, this is the exception rather than the norm.
16/17
So why is this? People teach subjects the way they learned them. There's a huge void in the literature that approaches materials in such a memorable way.
Let's change things, and make math more inclusive and accessible! For more materials, see:
math.clemson.edu/~macaule/class…
17/17
Let's change things, and make math more inclusive and accessible! For more materials, see:
math.clemson.edu/~macaule/class…
17/17
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