Group theory and generalizations Definition

P-group

Definition

Let $p$ be a prime number. A group $G$ is a $p$ -group when, for each $g \in G$ , the order of $g$ is $p^n$ for some integer $n \geq 0$ .

Remarks

A finite group is a $p$ -group if and only if $|G| = p^k$ for some $k \geq 0$ . One direction uses Lagrange's theorem, and the other direction uses Cauchy's theorem. The dihedral group $D_4$ of order $8$ is a $2$ -group that is not abelian. The trivial group is a $p$ -group for each prime $p$ .

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Group concept
Laws of exponents in a group statement
Order of an element definition
P-group definition

Tag	`06DAH3VS6M`
URI	`m2://06DAH3VS6M`
Kind	definition
Writing	W1 stated
MSC	20D15
Defines	P-group
Uses	Order of an element

P-group

Definition

Remarks

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