Normal subgroups

Let $H\subset G$ be a subgroup,

• $\forall g\in G$, $Hg = gH \iff gHg^{-1} = H \iff g^{-1}Hg = H \iff H$ is fixed by all inner automorphisms of $G \implies H$ is a normal subgroup of $G$. if $H \le G$ is normal, we denote $H \lhd G$
• $\forall g\in G, gHg^{-1} = H \iff gHg^{-1} \subseteq H$

Normal subgroup Test

A subgroup $H$ of a group $G$ is normal if and only if $xHx^{-1}\subseteq H$ for all $x\in G$

Normal group properties

• any subgroup in abelian group is a normal subgroup
• any center of group is a normal subgroup
• $G$ has unique subgroup $H$ with particular order $\implies H$ is normal. $gHg^{-1}$ is a subgroup of same order

Let $G$ be a group, and let $H$ be a subgroup. The normalizer of $H$ is a set $N_G(H) = N(H) = \{g\in G: gHg^{-1} = H\}$

Let $H\subset G$ be a subgroup,

• The normalizer is a subgroup of $G$
• $H\subseteq N(H)$
• $H$ is a normal subgroup of $G\iff N(H) = G$
• $H$ is a normal subgroup of $N(G)$

Quotient (Factor) Groups

Let $H\subset G$ be a normal subgroup, let's denote $G/H = \{gH:g\in G\}$ and a binary operation $\cdot : G/H \times G/H\to G/H$ as $aH\cdot bH = (ab)H$. The $(G/H, \cdot, eH = H)$ is the quotient group of G by H. (i.e. the set of left (or right) cosets of $H$ in $G$ is a group)

Quotient Groups Properties:

• any quotient of a cyclic group is cyclic
• any quotient of an abelian group is abelian

Theorem: $G/Z$

Let $G$ be a group with the center $Z(G)$

• $G/Z(G)$ is cyclic $\implies G$ is abelian
• $G/Z(G) \cong Inn(G)$

It also work if apply on the $H \le Z(G)$ with $G/H$ is the quotient group.

Cauchy's Theorem for Abelian Groups

Let $G$ be a finite Abelian group and let $p$ be a prime divisor of $|G|$. Then there exists an element $a\in G$ such that $|a| = p$.

Let $aH\in G/H, o_{G/H}(aH)$ is the smallest positive integer $k, s.t. a^k \in H$

$G$ is a finite abelian group and $p$ is a prime divisor of $|G| \implies \exists a\in G, o(a) = p$

• actually holds for any finite group (the first Sylow theorem)