Group Theory

[kanji]

ช่วยพิสูจน์หน่อยครับ

1. Let $G$ be a nontrivial group. Then $G$ is a finite $p-$group if and only if $|G|=p^k$ for some positive integer $k$.

[nooonuii]

Suppose G is a nontrivial finite p - group. Let $q$ be a prime divisor of $|G|$. By Cauchy's Theorem, $G$ has an element of order $q$. Since every element of $G$ has order $p^k$ for some $k\in N$, we conclude that $q = p$. This means that the only prime divisor of $|G|$ is $p$. Therefore, $|G| = p^n$ for some $n\in N$.

The converse is easy by Lagrange's Theorem.

[kanji]

ขอบคุณคุณnooonuii สำหรับข้อ 1 ครับ

2 Let $G$ be a finite group and $H$ be a Sylow p-subgroup of $G$. Then $H$ is a unique

Sylow p-subgroup of $G$ if and only if $H$ is a normal subgroup of $G$.

[nooonuii]

2. Hint:

$(\Rightarrow )$ Show that every conjugate of a Sylow p-subgroup is also a Sylow p-subgroup of $G$. Deduce that $H$ is normal in $G$ by using problem hypothesis.

$(\Leftarrow)$ By Sylow's Theorem, all Sylow p - subgroups are conjugate in $G$. Deduce that $H$ is a unique Sylow p-subgroup of $G$ by using problem hypothesis.