# Algebra Qual Prep Week 1: Groups Warmup

Tags: #qual_algebra

# Week 1: Finite Groups

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

## 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 $${\sharp}HK = {\sharp}H {\sharp}K / {\sharp}(H \cap K) = p\cdot p/1 = p^2$$

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

## Qual Problems

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