A group is a set $G$ together with a binary operation $\ast$ such that:

Closure: The set is closed under the binary operation. For all $a,b\in G,a\ast b\in G$.

Associativity: The binary operation is associative on the set. For all $a,b,c\in G,(a\ast b)\ast c=a\ast (b\ast c)$.

Identity: The set contains an identity element, denoted $e$. For all $a\in G,a\ast e=a$.

Inverses: All elements in the set have inverse elements in the set, denoted using $a^{-1}$. For all $a\in G$ there exists $a^{-1}\in G$ such that $a\ast a^{-1}=e$.

This set/operation combination $G$ is commonly denoted as the pair $(G,\ast )$.

Abelian groups are groups whose operation is commutative. For $a,b\in G,a\ast b=b\ast a$.

A subgroup $H$ of a group $G$ is a subset of $G$ together with the same operation as $G$ that still forms a group. The identity element of $G$ must also be the identity element of $H$.

A subgroup $H$ of a group $G$ is normal if its left and right cosets coincide, that is, if $gH=Hg$, i.e. $gH{g}^{-1}=H$, for all $g\in G$.

A cyclic group is a group where there exists some $g\in G$ such that every element in $G$ can be generated from the group operation applied to $g$.

That is, $G=\{g^n|n\in \mathbb{Z}\}$ when we think of the operation as multiplication, or $G=\{ng|n\in \mathbb{Z}\}$ when we think of the operation as addition.

The order of a finite group is the number of its elements. The order of group $G$ is denoted as $\mathrm{ord}(G)$ or $\Vert G\Vert$. The order of an element $a$ (also called period length or period) is the number of elements in the subgroup generated by $a$, and is denoted by $\mathrm{ord}(a)$ or $\Vert a\Vert$.

Any subgroup of a cyclic group is also cyclic - a cyclic subgroup.

A permutation is a bijection $\varphi :A\to A$, that is, a bijection from a set onto itself.

When $A$ is the finite set $\{1,2,3,\cdots ,n\},$ the group of all permutations of $A$ is the symmetric group on $n$ letters, and is denoted by $S_n$. It has $n!$ elements.

Permutations can be "multiplied", which is just composition. So if we have two permutations on $A$, called $\sigma$ and $\tau$, $\sigma \tau$ means their multiplication, and we apply from right to left, so $\tau$ applies first and then $\sigma$. The result will also be a permutation in $A$.

Let $\sigma$ be a permutation of $A$, the orbit of $\sigma$ containing $a\in A$ is the set

$$\{\cdots ,{\sigma }^{-2}(a),{\sigma }^{-1}(a),a,\sigma (a),{\sigma }^2(a),\cdots \}=\{{\sigma }^n(a)|n\in \mathbb{Z}\}.$$

So, $a,b\in A$ are in the same orbit of $\sigma$, written as $a\sim b$, if and only if $b={\sigma }^n(a)$ for some $n\in \mathbb{Z}$.

The relation "~" is an equivalence relation (reflexive, symmetric, transitive). This means $A$ can be partitioned into orbits of $\sigma$.

Let $\sigma$ be a permutation of $A$. The equivalence classes in $A$ determined by the equivalence relation "~" are the orbits of $\sigma$.

A permutation $\sigma \in S_n$ is a cycle if it has at most one orbit containing more than one element. The length of a cycle is the number of elements in its largest orbit.

A cycle in $S_n$ may be written as $(a_1,a_2,\cdots ,a_k)$ - this is called cyclic notation. It represents the permutation that sends $a_1\to a_2\to \cdots \to a_k\to a_1$.

Two or more cycles are disjoint if no element appears in more than one cycle.

A cycle of length 2 is a transposition.

A permutation of a finite set is even if it is the product of an even number of transpositions.

A permutation of a finite set is odd if it is the product of an odd number of transpositions.

The subgroup of $S_n$ consisting of all even permutations of $n$ letters is the alternating group $A_n$ on $n$ letters. If $n\ge 2$, then this set forms a subgroup of $S_n$ of order $n!/2$.

Let $H$ be a subgroup of $G$. Given $a\in G$, the subset $aH=\{ah|h\in H\}$ of $G$ is the left coset of $H$ containing $a$, while the subset $Ha=\{ha|h\in H\}$ is the right coset of $H$ containing $a$.

The number of left cosets of a subgroup $H$ in a group $G$ is the index $(G:H)$ of $H$ in $G$.

A homomorphism is a map $\varphi :G\to G^{\prime }$ between groups (not necessarily a bijection), $⟨G,\ast ⟩$ and $⟨G^{\prime },{\ast }^{\prime }⟩,$ that satisfies the homomorphism property:

$$\varphi (a\ast b)=\varphi (a){\ast }^{\prime }\varphi (b),\mathrm{\forall }a,b\in G.$$

The kernel of a homomorphism $\varphi$ is the set of elements that $\varphi$ sends to $e^{\prime }$, and it is denoted by $\ker (\varphi )$. It is a normal subgroup of $G$.

A group $G$ is simple if it has no proper nontrivial normal subgroups, that is, if and the only normal subgroups of $G$ are $\{e\}$ and $G$ itself.

The center of a group $G$ is all the elements that commute with all elements of $G$:

$$Z(G)=\{z\in G|zg=gz\text{ for all }g\in G\}.$$

The commutator subgroup of $G$ is the group $C$ generated by all elements of the set

$$\{ab{a}^{-1}{b}^{-1}|a,b\in G\}.$$