Useful Propositions of Group Theory

(This is a work in progess...)

Rearrangement Theorem:

$R\bf{G} = \bf{G}$  $(\forall R \in \bf{G})$.

Equivalent statement about the columns being $\bf{G} R = \bf{G}$  $(\forall R \in \bf{G})$

Each row and each column in the group multiplication table lists each of the group elements once and only once.
From this, it follows that no two elements may be in the identical location in two rows or two columns.
Thus, each row and each column is a rearranged list of the group elements.

Stated otherwise, given a group of n distinct elements $(I,a,b,c,\dots,n)$, the set of products $(aI,a^2,ab,ac,\dots,an)$ reproduces the $n$ original distinct elements in a new order.

Cayley's theorem:

Every group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$.

Cyclic Group

A set of elements $\{R, R^2, ..., R^n(=E)\}$ is called a cyclic group of order $n$.

A group $G$ is called cyclic if there exists an element $g$ in $G$ such that $G = \langle g \rangle = \{g^n: n\text{ is an integer}\}$

Proposition about $(\mathbb{Z}/n\mathbb{Z})^\times$ being cyclic:

$(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if $\varphi(n)=\lambda(n)$.
where $\varphi(n)$ is Euler's totient function and $\lambda(n)$ is Carmichael function.

$\varphi(n)$ = number of integers between $1$ & $n$ which are relatively prime to n.

$\lambda(n)$ = minimal raised power when all relatively prime integers raised to that number ($\lambda(n)$) modulus $n$ equal $1$

Or alternatively:
$(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if $n=1,2,4$, any power of an odd prime ($p^k$) or twice any power of an odd prime ($2p^k$).

Lagrange’s Theorem:

$H\le G\implies |H|$ divides $|G|$

Coset

Let $H = {A_1, A_2, \dots, A_n}$, where $H\subset G$.

Suppose $R_1, R_2, \dots \in G$ not contained in $H$.

Then the collection defined by $\bf{HR_k = {A_1R_k, A_2R_k,\dots,A_nR_k}}$ is called right coset of $H$ w.r.t to $R_k$

Conjugate

An element $B$ is said to be conjugate to $A$ w.r.t $R$ if $B = RAR^{-1}$

Normal Subgroup and Factor Group (Quotient Group):

$N\le G$ is normal if $g\cdot N\cdot g^{-1}=N$, $\forall g\in G$

OR

$N\triangleleft G\,\Leftrightarrow \;\forall \;n\in N,\;\forall \ g\in G\colon \;gng^{-1}\in N$

The key thing about Normal subgroups is that you can use $N$ to split $G$ into a bunch of cosets, and you can treat these cosets as elements in a group, called the Factor Group (Quotient Group)

In this factor group, the subgroup $N$ is the identity element