Tags: #qual_algebra

Week 1: Finite Groups

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Week 1 Topics

• Subgroups
  • The one-step subgroup test
    • \((x,y\in H\implies xy^{-1}\in H) \implies H\leq G\)
  • Cosets > \(xH := \left\{{xh{~\mathrel{\Big\vert}~}h\in H}\right\}, G/H := {\textstyle\coprod}_{x} xH\)
  • The index of a subgroup
  • Normal subgroups
  • Quotients
  • The normalizer of a subgroup
  • Maximal and proper subgroups
  • Characteristic subgroup
• Cauchy’s theorem
• Lagrange’s theorem
• Definitions and properties of common special families of groups:
  • Cyclic groups \(C_n\)
  • Symmetric groups \(S_n\)
  • Alternating groups \(A_n\)
  • Dihedral groups \(D_{n}\)
  • The quaternion group \(Q_8\)
  • Matrix groups \(\operatorname{GL}_n(k), {\operatorname{O}}_n(k), {\operatorname{SL}}_n(k), {\operatorname{SO}}_n(k)\)
  • \(p{\hbox{-}}\)groups
  • Free groups \(F_n\) (and presentations/relations)
• The 4 fundamental isomorphism theorems
• Finite groups of order \({\sharp}G \leq 20\)
• Structure:
  • Cyclic -> Abelian -> Nilpotent -> Solvable -> All Groups

Review Exercises

• State the definitions of the following:
  • Group morphism (aka group homomorphism)
  • Centralizer
  • Normalizer
  • Conjugacy class
  • Center
  • Inner automorphism
  • Commutator
  • \(p{\hbox{-}}\)group
• Write definitions or presentations for all of the special families of groups appearing above.
• State what it means for a cycle to be even or odd.
• Find a counterexample for the converse of Lagrange’s theorem.
• State the 4 fundamental isomorphism theorems

Unsorted Questions

For everything that follows, assume \(G\) is a finite group.

• \(H\leq G\) denotes that \(H\) is a subgroup of \(G\).
• \({\sharp}G\) denotes the order of \(G\).
• \(e\) or \(e_G\) denotes the identity element of \(G\).
• Multiplicative notation is generally used everywhere to denote the (possibly noncommutative) binary operation
• \(G/H\) is the set of left cosets of \(G\) by \(H\).

Cosets

• Let \(H\leq G\) be a subgroup (not necessarily normal). Prove that any two cosets \(xH, yH\in G/H\) have the same cardinality.

  Define a map \(m_g: G\to G\) where \(x\mapsto gx\), restrict to \(m_h:H\twoheadrightarrow gH\), inverse \((m_g)^{-1}= m_{g^{-1}}\)

• Prove the fundamental theorem of cosets: for \(xH, yH\in G/H\), \begin{align*} xH = yH \iff x^{-1}y\in H \iff y^{-1}x \in H \end{align*}

Use that \(xH = yH\iff x\sim y\) is an equivalence relation (reflexive/symmetric/transitive) - Suppose \({\sharp}G = pq\) with \(p, q\geq 2\) prime, and let \(H\leq G\) be a proper subgroup. Prove that \(H\) must be cyclic.

Use (and prove) the classification of groups of order \(p\).

Orders

• Prove Lagrange’s theorem.

Use \(G = {\textstyle\coprod}_{i=1}^n g_i H\), that cosets all have cardinality \({\sharp}H\), and \({\sharp}{\textstyle\coprod}X_i = \sum {\sharp}X_i\)

• Prove Cauchy’s theorem.

Induce on \({\sharp}G\). Assume \({\sharp}G > p\) and pick \(g\neq 1\). If \(p\divides {\sharp}g\), use cyclic group theory, so assume otherwise. Use that \(\sizeG = \sizeG/N \sizeN\) so \(p\) divides \({\sharp}G/N\), apply IH to get an element of order \(p\) in the quotient. Then \(y\not\in N\) but \(y^p\in N\), so \(\left\langle{y}\right\rangle\neq \left\langle{y^p}\right\rangle\) since \(y^p\in N \implies \left\langle{y^p}\right\rangle \subseteq N\). Get \(p\divides {\sharp}\left\langle{y}\right\rangle\), apply IH. - Prove that if \({\sharp}G\) is prime, then \(G\) is cyclic

> Assume there are two distinct generators and reach a contradiction.

• Prove that for every \(g\in G\), the order of \(g\) divides the order of \(G\).
• Prove that if \({\sharp}G = n\), then \(g^n = e\) for every \(g\in G\)

Normal Subgroups

• Let \(s\in G\), and state the definition of the centralizer of \(C_G(s)\) of \(s\) in \(G\).

  • Show that \(C(s) \leq G\) is a subgroup.
  • Let \(\left\langle{ s }\right\rangle \subseteq C_G(s)\), where \(\left\langle{ s }\right\rangle\) is the subgroup of \(G\) generated by \(s\).
  • Prove that \(\left\langle{ s }\right\rangle{~\trianglelefteq~}G\) is in fact a normal subgroup.

• Let \(H\leq G\) be a subgroup and \(N{~\trianglelefteq~}G\) be a normal subgroup. Show that \(NH \leq G\) is a subgroup.

• Let \(G_1, G_2\) be groups and \(H_2 \leq G_2\) a subgroup. Suppose \(\phi: G_1\to G_2\) is a group morphism.

  • Show that the image \(\phi(G_1) \leq G_2\) is a subgroup of \(G_2\)
  • Show that the preimage \(\phi^{-1}(H_2) \leq G_1\) is a subgroup of \(G_1\),
  • Show that the kernel \(\ker \phi {~\trianglelefteq~}G_1\) is a normal subgroup of \(G_1\).
  • Prove that group morphisms preserve coset structure in the following sense: \begin{align*} xH_1 = yH_1 \iff \phi(x)H_2 = \phi(y)H_2 .\end{align*}
  • Prove the first isomorphism theorem: \(\phi\) is injective \(\iff \ker \phi = \left\{{ e_{G_1} }\right\}\).

Symmetric Groups

• Let \(\sigma = (4\, 2\, 1)(6\, 1\, 3\, 2) \in S_6\) in cycle notation.
  • Write \(\sigma\) as a product of disjoint cycles.
  • Compute the order of \(\sigma\). What is the general theorem about the order of cycles?
  • Determine if \(\sigma\) is even or odd. What is the general theorem?
• Suppose \(\phi: S_n \to G\) with \(n\) even and \({\sharp}G = m\) odd.
  • Prove that if \(\tau \in S_n\) is a transposition, then \(\tau \in \ker \phi\).
  • Prove that in fact every \(\sigma \in S_n\) satisfies \(\sigma \in \ker \phi\), so \(\phi\) is the trivial morphism.
  • Does this hold if \(n\) is odd?

Matrix Groups

• Let \({\mathbb{F}}_p\) be the finite field with \(p\) elements, where \(p\) is a prime. Show that the centers of \(\operatorname{GL}_n({\mathbb{F}}_p)\) and \({\operatorname{SL}}_n({\mathbb{F}}_p)\) consist only of scalar matrices.
  • Show that the scalars \(\zeta\) that appear in scalar matrices \(Z({\operatorname{SL}}_n({\mathbb{F}}_p))\) are roots of unity in \({\mathbb{F}}_p\), i.e. \(\zeta^p = 1\).
• Determine the orders \({\sharp}\operatorname{GL}_n({\mathbb{F}}_p)\) and \({\sharp}{\operatorname{SL}}_n({\mathbb{F}}_p)\).

Warmup Problems

• (Important) Prove that if \(G/Z(G)\) is cyclic then \(G\) is abelian.

  Write \(Z = Z(G)\), fix \(x,y\in G\). Since \(G/Z = \left\langle{gZ}\right\rangle\), \(xZ = (gZ)^m = g^mZ\) and \(yz = (gZ)^n = g^nz\) \(g^{-m}x, g^{-n}y \in Z \implies x = g^m z_1, y = g^n z_2\) \(xy = g^m z_1 g^n z_2\), everything commutes.

• (Important) Classify all groups of order \(p^2\).

  Must be abelian since quotient is cyclic. If there’s an element of order \(p^2\), cyclic, done. Else every element \(a\neq 1\) must have order \(p\). Then \(\left\langle{a}\right\rangle\neq G\), so pick \(b\) in its complement, it has order \(p\). Call these two subgroups \(H, K\) Recognize direct products: abelian implies both are normal, \(H \cap K = \left\{{1}\right\}\). and \(\sizeHK = {\sharp}H {\sharp}K / {\sharp}(H \cap K) = p\cdot p/1 = p^2\)

• (Important) Show that if \(H\leq G\) and \([G: H] = 2\) then \(H\) is normal.

  Index 2 implies partition into 2 left cosets: \(H, gH\), or two right cosets \(H, Hg'\) Note that \(gH = G\setminus H = Hg'\) Pick \(x\), want to show that \(xHx^{-1}= H\), so \(xH = Hx\). Case 1: \(x\in H\implies xH = H = Hx\) Case 2: \(xH\neq H \implies xH = gH\). Similarly \(Hx \neq H \implies Hx = Hg'\), so \begin{align*}xH = gH = G\setminus H = Hg' = Hx\end{align*}

  • Suppose that the same result holds with 2 replaced by \(p\) defined as the smallest prime factor of \({\sharp}G\)

• 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 \(X\).

• Let \(G\leq H\) where \(H\) is a finite \(p{\hbox{-}}\)group, and suppose \(\phi: G\to H / [H, H]\) be defined by composing the inclusion \(G\hookrightarrow H\) with the natural quotient map \(H \to H/[H, H]\).

  Prove that \(G= H\) by induction on \({\sharp}H\) in the following way:

  • Letting \(N{~\trianglelefteq~}H\) be any nontrivial normal subgroup of \(H\), use the inductive hypothesis to show that \(H = GN\).
  • Let \(Z = Z(H)\) be the center of \(H\). Using that \(GZ = H\) by (1), show that \(G \cap Z \neq \emptyset\). Set \(N \coloneqq G \cap Z\) and apply (1) to conclude.

• Determine all pairs \(n, p\in {\mathbb{Z}}^{\geq 1}\) such that \({\operatorname{SL}}_n({\mathbb{F}}_p)\) is solvable.

Qual Problems