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 \(\operatorname{GL}_n({\mathbb{F}}_p)\)?

• Identify \(\mathop{\mathrm{Aut}}({\mathbb{Z}}/p)\) and \(\mathop{\mathrm{Aut}}(\prod_{i=1}^n {\mathbb{Z}}/p)\) for \(p\) a prime.

  • Identify \(\mathop{\mathrm{Aut}}({\mathbb{Z}}/n)\) for \(n\) composite.

• How many elements in \(\mathop{\mathrm{Aut}}({\mathbb{Z}}/20)\) 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 \(n, p\in {\mathbb{Z}}^{\geq 1}\) such that \({\operatorname{SL}}_n({\mathbb{F}}_p)\) 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: