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?

Hint: consider the derived series.

Qual Problems

Needs some Sylow theory: