Tags: #qualifying_exam #active_projects #qual_algebra

Week 2: Finite Groups

Topics

Recognition of direct products and semidirect products
Amalgam size lemma: \({\sharp}HK = {\sharp}H {\sharp}K / {\sharp}(H\cap K)\)
Group actions
- Orbit-stabilizer
- The class equation,
- Burnside’s formula
- Important actions
  - Self-action by left translation (the left-regular action)
    - The assignment \(g\mapsto \psi_g\in \operatorname{Sym}(G)\) where \(\psi_g(x) \coloneqq gx\) is sometimes referred to as the Cayley representation in qual questions, or sometimes a permutation representation since \(\operatorname{Sym}(G) \cong S_n\) as sets where \(n\coloneqq{\sharp}G\)
    - See the Strong Cayley Theorem
  - Self-action by conjugation
  - Action on subgroup lattice by left-translation
  - Action on cosets of a fixed \(G/H\) by left-translation
Transitive subgroups
- How these are related to Galois groups
FTFGAG: The Fundamental Theorem of Finitely Generated Abelian Groups
- Invariant factors
- Elementary divisors
Simple groups
Automorphisms
- Inner automorphisms
- Outer automorphisms (not often tested directly)
- Characteristic subgroups (not often tested directly)
Series of groups (not often tested)
- Normal series
- Central series
- The Jordan-Holder theorem
  - Composition series
- Solvable groups
  - Derived series
- Nilpotent groups
  - Lower central series
  - Upper central series

A remark: automorphisms and series of groups aren’t often directly tested on the qual, but are useful practice. Simple/solvable groups do come up often.

Exercises

Warmup

Show that if \(H, K \leq G\) are subgroups and \(H \in N_G(H)\), then \(HK\) is a subgroup.
- Find a counterexample where \(H\leq G\), \(K\) is only a subset and not a subgroup, and \(HK\) fails to be a subgroup?
Prove the “Recognizing direct products” theorem: if \(H, K\) are normal in \(G\) with \(H \cap K = \emptyset\) and \(HK = G\), then \(G\cong H\times K\).
- Hint: write down a map \(H\times K\to G\) and follow your nose!
- How can you generalize this to 3 or more subgroups?
State definitions of the following:
- Group action
- Orbit
- Stabilizer
- Fixed points
State the orbit-stabilizer theorem
State the class equation. Can you derive this from orbit-stabilizer?
Show that the center of a \(p{\hbox{-}}\)group is nontrivial
Important: Pick your favorite composite number \(m = \prod p_i^{e_i}\) and classify all abelian groups of that order.
- Write their invariant factor decompositions and their elementary divisor decompositions. Come up with an algorithm for converting back and forth between these.
Prove that if \(H\leq G\) is a proper subgroup, then \(G\) can not be written as a union of conjugates of \(H\). - Use this to prove that if \(G = \operatorname{Sym}(X)\) is the group of permutations on a finite set \(X\) with \({\sharp}X = n\), then there exists a \(g\in G\) with no fixed points in \(X\).
Define what a composition series is, and state what it means for a group to be simple, solvable, or nilpotent.
- How are the derived and lower/upper central series defined? What type(s) of the groups above does each series correspond to?

Group Actions

For each of the following group actions, identify what the orbits, stabilizers, and fixed points are. If possible, describe the kernel of each action, and its image in \(\operatorname{Sym}(X)\).
- \(G\) acting on \(X=G\) by left-translation: \begin{align*}g\cdot x := gx\end{align*} .
- \(G\) acting on \(X=G\) by conjugation: \begin{align*}g\cdot x := gxg^{-1}\end{align*}
- \(G\) acting on its set of subgroups \(X:=\left\{{H{~\mathrel{\Big\vert}~}H\leq G}\right\}\) by conjugation: \begin{align*}g\cdot H := gHg^{-1}\end{align*}
- For a fixed subgroup \(H\leq G\), \(G\) acting on the set of cosets \(X := G/H\) by left-translation: \begin{align*}g\cdot xH := (gx)H\end{align*}
Suppose \(X\) is a \(G{\hbox{-}}\)set, so there is a permutation action of \(G\) on \(X\). Let \(x_1, x_2\in X\), and show that the stabilizer subgroups \({\operatorname{Stab}}_G(x_1), {\operatorname{Stab}}_G(x_2)\leq G\) are conjugate in \(G\).
Let \([G:H] = p\) be the smallest prime dividing the order of \(G\). Show that \(H\) must be normal in \(G\).
Show that if \(G\) is an infinite simple group, then \(G\) can not have a subgroup of finite index.

Hint: use the left-regular action on cosets.
Show that every subgroup of order 5 in \(S_5\) is a transitive subgroup.

Automorphisms

How do you compute the totient \(\phi(p)\) for \(p\) prime? Or \(\phi(n)\) for \(n\) composite?
What is the order of \(\operatorname{GL}_n({ \mathbf{F} }_p)\)?
Identify \(\mathop{\mathrm{Aut}}({\mathbf{Z}}/p)\) and \(\mathop{\mathrm{Aut}}(\prod_{i=1}^n {\mathbf{Z}}/p)\) for \(p\) a prime.
- Identify \(\mathop{\mathrm{Aut}}({\mathbf{Z}}/n)\) for \(n\) composite.
How many elements in \(\mathop{\mathrm{Aut}}({\mathbf{Z}}/20)\) have order 4?
Find two groups \(G\not\cong H\) where \(\mathop{\mathrm{Aut}}G\cong \mathop{\mathrm{Aut}}H\).
Let \(H, K \leq G\) be subgroups with \(H\cong K\). Is it true that \(G/H \cong G/K\)?

Hint: consider a group with distinct subgroups of order 2 whose quotients have order 4.
Show that inner automorphisms send conjugate subgroups to conjugate subgroups.
Show that for \(n\neq 6\), \(\mathop{\mathrm{Aut}}(S_n) = \mathop{\mathrm{Inn}}(S^n)\).

Series of Groups

Determine all pairs \(n, p\in {\mathbf{Z}}^{\geq 1}\) such that \({\operatorname{SL}}_n({ \mathbf{F} }_p)\) is solvable.
If \({\sharp}G = pq\), is \(G\) necessarily nilpotent?

Hint: consider \(Z(S_3)\).
Show that if \(G\) is solvable, then \(G\) contains a nontrivial normal subroup.
- What does this mean on the Galois theory side?