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
View other prime-parametrized group properties | View other group properties

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.