Tags: #qual_algebra
Week 1: Finite Groups
Week 1 Topics

Subgroups

The onestep 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

The onestep subgroup test
 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.