definition of p-group
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of group.
- The reader knows a definition of order of element of group.
Target Context
- The reader will have a definition of p-group.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*G\): \(\in \{\text{ the groups }\}\)
\( p\): \(\in \{\text{ the prime numbers }\}\)
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Conditions:
\(\forall g \in G (\exists n \in \mathbb{N} (\vert (g) \vert = p^n))\)
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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 \(\vert G \vert = p^n\).
But this definition is valid for also any infinite group.
