Sylow’s theorems and their Applications

Sylow’s theorems and their applications are major sub-unit of the forthcoming entrance exams like CSIR-NET and GATE. This major topic carries a good amount of marks in these exams and also improved your rankings all over.

In this article, we’ll try to understand the three of Sylow’s theorem and their application as simple we can. Here are the GATE notes and if you are looking for CSIR NET Maths Notes for Group Theory, you must check this article below. You’ll also get the skill to how to find number of sylow p-subgroups.

Sylow’s theorems and their Applications – CSIR-NET & GATE Notes

p-group: A group is said to be p-group if $O(G)=p$ where $p$ is prime.
Example Given: Let $O(G)=49=7^{2}$ then $G$ is 7 – Group

Cauchy Theorem: If G has a finite abelian group and $P \mid O(G)$ then $G$ has a subgroup of order $p$ and which is isomorphic to $\mathbb{Z}_{p}$ .

Remark: If $G$ be a finite group and $p \mid \mathrm{O}(G)$ then $G$ has elements of order $P$ .

All 3 Theorems of Sylow’s:

Sylow 1st Theorem: If G be a finite group and $p^{n}|0(G)$ then G has subgroup of order $p^{n}$ . This theorem helps you to find how to find Sylow p-subgroups orders.

Sylow 2nd Theorem: Any two $p$ -sylow subgroup of $G$ are conjugate. i.e Let $H_{1}$ and $H_{2}$ are two $p$ – sylow subgroup of G then $\exists x \in G$ such that $H_{1}=x{H}_{2} x^{-1}$ .

The 3rd Sylow’s Theorem is the key to how to find number of sylow p-subgroups.

Sylow – 3rd Theorem: The number of $p-\operatorname{ssG}(n)$ in G is equal to $1+kp$ such that $1+kp \mid o(G)$ and $k=0,1,2,...$ .

Some Important Notes on Sylow’s Theorems:

If $H$ is unique $p-\operatorname{SSG}$ in G if and only if $H \mathrel{\unlhd} G$ .
Let $G=GL\left(n, F_{q}\right)$ . then $q-SSG$ are also $q^{\frac{n(n-1)}{2}}$ .
Let $G=SL\left(n, Z_{p}\right)$ . then $p-SSG$ are also $p^{\frac{n(n-1)}{2}}$ .

Here I’m sharing my notes for Sylow’s Theorem from direct copy.

Applications of Sylow Theorem

Sylow’s First Theorem Application: This theorem is generally applied when we have given the order of group G, then we can find the order of the subgroup of G.

Sylow’s 1st Theorem Problem: Let $o(G)=24=8 \times 3$ .
Find the order of subgroups in group G. (How to find sylow p-subgroups)
Solution: Here $o(G)=24=8 \times 3 = {2}^{3} \times 3$ $2 \mid \mathrm{o}(\mathrm{G})$ then $\mathrm{G}$ has a subgroup of order 2.
$3 \mid \mathrm{o}(\mathrm{G})$ then $\mathrm{G}$ has a subgroup of order 3.
$2^{2} \mid \mathrm{o}(\mathrm{G})$ then $\mathrm{G}$ has a subgroup of order 4.
$2^{3} \mid \mathrm{o}(\mathrm{G})$ then $\mathrm{G}$ has a subgroup of order 8.

The second theorem is applied to find the conjugate subgroups of a given group G. And Sylow’s third theorem is applied to find the number of p-SSG in a group G (how to find number of sylow p-subgroups in a group G).

Hope you like this post on the GATE and NET important topic Sylow’s Theorem and the application. The problems are basically fact-based and comparatively easy than the proofs we learned in colleges.