Tags: #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
 Orbitstabilizer
 The class equation,
 Burnside’s formula

Important actions

Selfaction by left translation (the leftregular 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
 Selfaction by conjugation
 Action on subgroup lattice by lefttranslation
 Action on cosets of a fixed \(G/H\) by lefttranslation

Selfaction by left translation (the leftregular action)

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 JordanHolder 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 orbitstabilizer theorem
 State the class equation. Can you derive this from orbitstabilizer?
 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 lefttranslation: \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 lefttranslation: \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 leftregular 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?
Hint: consider the derived series.
Qual Problems
Needs some Sylow theory: