Note - Results about Finite Groups

notegroup-theory

For finite group $(G,\cdot )$ with $|G|<\infty$.

• If $|G|=p$ prime, then $G$ is cyclic, and $G\simeq ({\mathbb{Z}}_p,+)$.
• Lagrange: If $H\le G$, then $|G|=|H|\cdot [G:H]$ $\leftarrow$ $\#$ of left cosets of $H$ in $G$.
• Class Equation: $|G|=|Z(G)|+{\sum }_{i=1}^r\frac{|G|}{|Z(x_i)|}$ $Z(G)$ is the center of $G$, $x_1,\dots ,x_r$ are one element from each distinct conjugacy class with more than one element.
• Cauchy: If $p\ge 2$ prime and $p\mid |G|$, then there exists $x\in G$ such that ${\mathrm{ord}}_G(x)=p$.
• If $|G|=p^2$ and $p$ prime, then $(G,\cdot )$ is commutative.
• Structure Theorem for Finite Commutative Groups: If $(G,\cdot )$ commutative and finite, then $G\simeq$ product of cyclic p-groups $({\mathbb{Z}}_{p^k},+)$ for prime $p$ and $k\ge 1$.
• If $N⊴G$, $({}^G{/}_N,\cdot )$ is a finite group with $[G:N]=\frac{|G|}{|N|}$ elements.
• If $G$ is a p-group, then $Z(G)\ne \{e\}$ (also p-group).
• If ${}^G{/}_{Z(G)}$ is cyclic, then ${}^G{/}_{Z(G)}=\{e\}$, $G=Z(G)$ and $G$ is commutative.
• If $g^2=e\forall g\in G$, then $G$ is a commutative 2-group. Ex: groups with n elements.
  • $n=1$: $G=\{e\}$.
  • $n=2$: $G={\mathbb{Z}}_2$.
  • $n=3$: $G={\mathbb{Z}}_3$.
  • $n=4$: $G={\mathbb{Z}}_4$ or $G\simeq {\mathbb{Z}}_2\times {\mathbb{Z}}_2$.
  • …
• Sylow Theorems