Letters like $G$ will denote groups. The notation $H\le G$ means that $H$ is a subgroup of $G$.
Letters like $X$ will denote graphs. A graph $X = (V, E)$ will be undirected by definition. We will use the word digraph to refer to directed graphs
Given a set $\Omega$, the symmetric group $\sym(\Omega)$ is the group of all permutations of $\Omega$. The set $\Omega$ is the permutation domain or $G$-space.
Let $G$ be a group. A $G$-action on $\Omega$ is a homomorphims $G:\to \mathrm{Sym}(\Omega)$. For $\pi\in G$, and $a\in\Omega$, $a^{\pi}$ denotes the image of $a$ under $\pi$.
A subgroup $G\leq \mathrm{Sym}(\Omega)$ is a permutation group. When we are talking about a permutation group, we will often omit the action, and it will be understood we mean its natural action on $\Omega$.
The size $|\Omega|$ of the permutation domain is the degree of $G$. This is not to be confused with the order of $G$ which is $|G|$. So the order of a permutation group of degree $n$ is at most $n!$.Example: there is a map $G\to \mathrm{Sym(G)}$ given by conjugation: $a^g = g^{-1}ag$.
$S_n = \mathrm{Sym}([n])$, where $[n] = \{1, 2, \dots, n\}$.
A permutation $\sigma\in S_n$ can be specified by listing the image of every element of [n] or as the product of disjoint cycles. This is called the cycle decomposition of $\sigma$. Example: the permutation that takes $1\to 4\to 6\to 1$ and $2\to 7\to 2$ is represented as $(146)(27)$. We can write the same decomposition as $(72)(614)$.
The cycle type of a permutation $\sigma$ is the ordered list $(k_i, \dots, k_t)$, where $k_1\geq \dots \geq k_t$ are the lengths of the cycles in the cycle decomposition of $\sigma$.
For instance, the cycle type of $(72)(614)\in S_7$ is $(3,2,1,1)$.Exercise 2.1: $\sigma, \tau\in S_n$ are conjugates (i.e., $(\exists \rho)(\tau = \rho^{-1}\sigma\rho)$) if and only if they have the same cycle type.
Exercise 2.2: Let us pick a random permutation $\sigma$.
Consider a $G$-action $G\to \mathrm{Sym}(\Omega)$. The stabilizer of $a\in\Omega$ in $G$ is $G_a = \{\pi \in G \mid a^\pi = a\}$.
The orbit of $a$ in $G$ is $a^G = \{a^\pi \mid \pi \in G\}$. We call the quantity $|a^G|$ the length of the orbit.
Exercise 3.1: The orbits form a partition of the permutation domain.
Exercise 3.2: Let $G$ act on $\Omega$. Prove that the index of the stabilizer of $a\in\Omega$ is the length of the orbit of $a$. In notation: $(\forall a\in \Omega) (|G : G_a|=|a^G|)$.
Exercise 3.3. (a) ("Burnside's Lemma"): Let $G$ act on $\Omega$. Prove: the average number of fixed points of elements of of $G$ is equal to the number of orbits of $G$. (b) Prove: if $|\Omega|\ge 2$ and $G$ acts transitively on $\Omega$, then $G$ has a fixed-point-free element.
We say that a $G$-action is transitive if $(\forall a, b\in \Omega)(\exists\pi\in G)(a^\pi = b)$. In other words this means $a^G = \Omega$ (there is just one orbit).
A block of imprimitivity of an action of $G$ on $\Omega$ is a subset $\Delta \subseteq \Omega$ such that $(\forall \pi\in G) (\Delta^\pi =\Delta \text{ or } \Delta^\pi \cap \Delta = \emptyset)$. The trivial blocks are the singletons and $\Omega$.
A transitive $G$-action is primitive if the degree is $\ge 2$ and there are no nontrivial blocks.
Exercise 4.1: If $G$ is a primitive permutation group and $N\triangleleft G$, $N\neq 1$, then $N$ is transitive.
Exercise 4.2: A transitive $G$-action is primitive if and only if the stabilizer $G_a$ $(a\in\Omega)$ is a maximal subgroup.
The left and right regular representations of a group $G$, denoted by $\lambda$ and $\rho$, respectively, are the actions of $G$ on itself defined by $\lambda_g: x\mapsto g^{-1}x$, and $\rho_g: x\mapsto xg$.
Exercise 5.1: Prove: the blocks of imprimitivity of $\rho(G)$ are precisely the right cosets of subgroups of $G$.
Exercise 5.2: (a) Prove: $\rho(G)$ is the centralizer of $\lambda(G)$ in $\sym(G)$ and vice versa. (b) Prove that $\rho(G)\lambda(G)$ is a group of order $|G|^2/|Z(G)|$. ($Z(G)$ denotes the center of $G$). (c) Prove: $\rho(G)\lambda(G)$ is primitive if and only if $G$ is simple.
The action of $G$ on $\Omega$ is semiregular if $(\forall a\in\Omega)(G_a=1)$.
The action of $G$ on $\Omega$ is regular if it is semiregular and transitive.
Exercise 5.3: (a) The centralizer of a transitive permutation group in $\sym(\Omega)$ is semiregular. (b) The centralizer of a semiregular permutation group in $\sym(\Omega)$ is transitive. (c) The centralizer of a regular permutation group in $\sym(\Omega)$ is regular.
Exercise 5.4: If $G$ is a regular permutation group, then $|G| = |\Omega|$.
A permutational isomorphism of two permutation groups is a group isomorphism induced by a bijection of the permutation domains.
Exercise 5.5: $(\forall G)(\exists $ a unique regular permutation group isomorphic to $G)$ (unique up to permutational isomorphism).
Corollary 5.6 $\rho(G)$ and $\lambda(G)$ are permutationally isomorphic.
Exercise 5.7 If $G$ is transitive and abelian, then it is regular.
Definition $H\leq G$ is a characteristic subgroup if it is invariant under all automorphisms of $G$. We write $H\, \mathrm{char}\, G$.
Exercise 6.1 (a) Prove: the relation "is a normal subgroup of" is not transitive. Find the smallest group that gives a counterexample. (b) Prove: if $K \,\mathrm{char}\, H\triangleleft G$, then $K\triangleleft G$.
Recall $G$ is simple if it has order $\ge 2$ and no non-trivial normal subgroups.
Definition $G$ is charachteristically simple if it has no non-trivial characteristic subgroups.
Exercise 6.2 $G$ is characteristically simple if and only if $G\cong T\times\dots\times T$, for some simple group $T$.
Exercise 6.3 Assume $G\cong T\times\dots\times T$ ($k$ copies) for some simple group $T$. Prove: (a) If $T$ is nonabelian, then the number of normal subgroups of $G$ is $2^k$. (b) If $T$ is abelian (i.e., cyclic of prime order) and $k\ge 2$, then the number of normal subgroups of $G$ is greater than $2^k$. Determine the exact number. (Not a closed-form expression but a simple formula.)
Definition A minimal normal subgroup of $G$ is a normal subgroup $M\triangleleft G$ such that $M\neq 1$ and the only normal subgroups of $G$ contained in $M$ are $1$ an $M$. (Example: $G$ is a minimal normal subgroup of itself exactly if $G$ is simple.)
Exercise 7.1 Minimal normal subgroups are characteristically simple.
Exercise 7.2 Prove: the degree of a primitive solvable permutation group is always a prime power.
Exercise 7.3 If $G$ is a transitive group of degree $p^k$, then all Sylow $p$-subgroups of $G$ are transitive.
A transposition is a cycle of length $2$ in $S_n$. There are $\binom{n}{2}$ transpositions in $S_n$.
Exericse 8.1: (a) The transpositions generate $S_n$. (b) $S_n$ is generated by the $n-1$ transpositions that exchange neighbors, viz., $(1,2)$, $(2,3)$, ..., $(n-1,n)$.
(c) $S_n$ is generated by any tree of transpositions. (d) If a set of $k$ transpositions generates $S_n$, then $k\ge n-1$.The alternating group of degree $n$ is the subgroup $A_n\leq S_n$ of even permutations.
For $n\ge 2$ we have $|S_n : A_n|=2$.Exercise 8.2: The alternating group is generated by the $3$-cycles.
Exercise 8.3: Let $H\le G$. Prove: if $|G:H|=2$, then $(\forall x \in G)(x^2\in H)$.
Hint: prove that $H\triangleleft G$.Exercise 8.4: If $G\leq S_n$ and $|S_n:G| = 2$, then $G = A_n$.
Exercise 8.5: True or False: if $|G:H|=3$, then $(\forall x\in G) (x^3 \in H) $?
The center $Z(G)$ of a group $G$ is the set of those elements that commute with all elements, i.e., $Z(G)= \{g\in G \mid (\forall h\in G) (gh = hg) \}$.
Exercise 9.1: (a) For $n\geq 3$ we have $Z(S_n)=1$. (b) For $n\geq 4$ we have $Z(A_n)=1$. (c) Find the center of the dihedral group $D_n$ (the group of symmetries of the regular $n$-gon). (d) What is the center of the quaternion group of order 8?
Let $F$ be a field. Let $\gl(n,F)$ denote the group of all nonsingular $n\times n$ matrices over $F$ (the "general linear group").
Exercise 9.2: Prove: for $n\ge 2$, the center of $\gl(n,F)$ consists of the scalar matrices $\lambda I$ $(\lambda\in F^{\times} = F\setminus \{0\}$ and $I$ denotes the identity matrix).
$\varphi:G\to H$ is an isomorphism of $G$ and $H$ if it is a bijective homomorphism. We will denote the set of all isomorphisms by $\mathrm{ISO}(G, H)$.
An automorphism of $G$ is an isomorphism from $G$ to itself. We will denote the set of all automorphisms of $G$ by $\mathrm{Aut}(G)$. Note that $\mathrm{Aut}(G)\leq \mathrm{Sym}(G)$.
Exercise 10.1: If $|G| = n$, then $|\mathrm{Aut}(G)|\leq n^{1+\log_2 n}$.
Let $\rho:G\to \mathrm{Sym}(G)$ be the action of $G$ on itself by conjugation. Then the group of inner automorphisms of $G$ is $\mathrm{Inn}(G) := \mathrm{Im}(\rho)$.
Exercise 11.1: There is a natural homomorphism $G\to \mathrm{Inn}(G)$. What is the kernel of this homomorphism?
Exercise 11.2: Prove: $\mathrm{Inn}(G)\triangleleft\mathrm{Aut(G)}$.
The group of outer automorphisms of $G$ is $\mathrm{Out}(G):= \mathrm{Aut}(G)/\mathrm{Inn}(G)$.
Exericse: $|\mathrm{Out}(S_n)| = 1$, except for $n=6$, where it is $2$.