841: p-Group

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definition of p-group

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Topics

About: group

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of group.
• The reader knows a definition of order of element of group.

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Target Context

• The reader will have a definition of p-group.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$\ast G$: $\in \{\text{ the groups }\}$
$p$: $\in \{\text{ the prime numbers }\}$
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Conditions:
$\mathrm{\forall }g\in G(\mathrm{\exists }n\in \mathbb{N}(|(g)|=p^n))$
//

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2: Note

When any group, $G$, is finite, there is the proposition that $G$ is a $p$-group if and only if there is an $n\in \mathbb{N}$ such that $|G|=p^n$.

But this definition is valid for also any infinite group.

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References

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