# P-elementary group

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The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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## Definition

### Symbol-free definition

Let $p$ be a prime. A finite group is termed $p$-elementary if it is the direct product of a $p$-group and a cyclic group of order relatively prime to $p$.

A group is termed elementary if it is $p$-elementary for some prime $p$.

### Definition with symbols

Let $p$ be a prime. A finite group $G$ is termed $p$-elementary if $G$ is the internal direct product of a $p$-subgroup $P$ and a cyclic subgroup $C$ whose order is relatively prime to $p$.

A group is termed elementary if it is $p$-elementary for some prime $p$.

## Property theory

### Relation with other properties

Elementary groups are nilpotent. This is because cyclic groups are nilpotent, and $p$-groups are also nilpotent, and a product of nilpotent groups is nilpotent.