Math4302 Modern Algebra (Lecture 23)

Group

Group acting on a set

Theorem for the orbit of a set with prime power group

Suppose $X$ is a $G$-set, and $|G|=p^n$ where $p$ is prime, then $|X_G|\equiv |X|\mod p$.

Where $X_G=\{x\in X|g\cdot x=x\text{ for all }g\in G\}=\{x\in X|\text{orbit of }x\text{ is trivial}\}$

Corollary: Cauchy’s theorem

If $p$, where $p$ is a prime, divides $|G|$, then $G$ has a subgroup of order $p$. (equivalently, $g$ has an element of order $p$)

This does not hold when $p$ is not prime.

Consider $A_4$ with order $12$, and $A_4$ has no subgroup of order $6$.

Corollary: Center of prime power group is non-trivial

If $|G|=p^m$, then $Z(G)$ is non-trivial. ($Z(G)\neq \{e\}$)

Proposition: Prime square group is abelian

If $|G|=p^2$, where $p$ is a prime, then $G$ is abelian.

Classification of small order

Let $G$ be a group

• $|G|=1$
  • $G=\{e\}$
• $|G|=2$
  • $G\simeq\mathbb{Z}_2$ (prime order)
• $|G|=3$
  • $G\simeq\mathbb{Z}_3$ (prime order)
• $|G|=4$
  • $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2$
  • $G\simeq\mathbb{Z}_4$
• $|G|=5$
  • $G\simeq\mathbb{Z}_5$ (prime order)
• $|G|=6$
  • $G\simeq S_3$
  • $G\simeq\mathbb{Z}_3\times \mathbb{Z}_2\simeq \mathbb{Z}_6$

• $|G|=7$
  • $G\simeq\mathbb{Z}_7$ (prime order)
• $|G|=8$
  • $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$
  • $G\simeq\mathbb{Z}_4\times \mathbb{Z}_2$
  • $G\simeq\mathbb{Z}_8$
  • $G\simeq D_4$
  • $G\simeq$ quaternion group $\{e,i,j,k,-1,-i,-j,-k\}$ where $i^2=j^2=k^2=-1$, $(-1)^2=1$. $ij=l$, $jk=i$, $ki=j$, $ji=-k$, $kj=-i$, $ik=-j$.
• $|G|=9$
  • $G\simeq\mathbb{Z}_3\times \mathbb{Z}_3$
  • $G\simeq\mathbb{Z}_9$ (apply the corollary, $9=3^2$, these are all the possible cases)
• $|G|=10$
  • $G\simeq\mathbb{Z}_5\times \mathbb{Z}_2\simeq \mathbb{Z}_{10}$
  • $G\simeq D_5$
• $|G|=11$
  • $G\simeq\mathbb{Z}_11$ (prime order)
• $|G|=12$
  • $G\simeq\mathbb{Z}_3\times \mathbb{Z}_4$
  • $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_3$
  • $A_4$
  • $D_6\simeq S_3\times \mathbb{Z}_2$
  • ??? One more
• $|G|=13$
  • $G\simeq\mathbb{Z}_{13}$ (prime order)
• $|G|=14$
  • $G\simeq\mathbb{Z}_2\times \mathbb{Z}_7$
  • $G\simeq D_7$

Lemma for group of order $2p$ where $p$ is prime

If $p$ is prime, $p\neq 2$, and $|G|=2p$, then $G$ is either abelian $\simeq \mathbb{Z}_2\times \mathbb{Z}_p$ or $G\simeq D_p$