Group theory and generalizations Finite groups Definition

Sylow p-subgroup

Definition

Let $G$ be a finite group and let $p$ be a prime number. Write $|G| = p^n m$ with $n \geq 0$ and $p \nmid m$ . A Sylow $p$ -subgroup of $G$ is a subgroup of $G$ of order $p^n$ . The set of the Sylow $p$ -subgroups of $G$ is written $\operatorname{Syl}_p(G)$ , and its size is written $n_p$ .

Remarks

A Sylow $p$ -subgroup is a maximal p-subgroup of $G$ . If $p$ does not divide $|G|$ , then the only Sylow $p$ -subgroup is $\{e\}$ . In $S_3$ , the Sylow $3$ -subgroup is $A_3$ , and there are three Sylow $2$ -subgroups, each of order $2$ .

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Group concept
Laws of exponents in a group statement
Subgroup definition
Order of an element definition
P-group definition
Sylow p-subgroup definition

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Tag	`0JWZX6FKSH`
URI	`m2://0JWZX6FKSH`
Aliases	`sylow-subgroup`
Kind	definition
Writing	W0 stub
MSC	20D20
Defines	Sylow p-subgroup
Uses	P-group, Subgroup

Sylow p-subgroup

Definition

Remarks

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