Definition:Sylow p-Subgroup/Definition 3

Definition

Let $p$ be prime.

Let $G$ be a finite group whose order is denoted by $\order G$.

Let $n$ be the largest integer such that:

$p^n \divides \order G$

where $\divides$ denotes divisibility.

A Sylow $p$-subgroup is a $p$-subgroup of $G$ which has $p^n$ elements.

Also see

• Equivalence of Definitions of Sylow $p$-Subgroup

Source of Name

This entry was named for Peter Ludwig Mejdell Sylow.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.5$. Orbits: Example $121$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Exercise $25.18$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 56$. First Sylow Theorem