P-elementary group
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(Redirected from Elementary group)
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
Contents
Definition
Symbol-free definition
Let be a prime. A finite group is termed
-elementary if it is the direct product of a
-group and a cyclic group of order relatively prime to
.
A group is termed elementary if it is -elementary for some prime
.
Definition with symbols
Let be a prime. A finite group
is termed
-elementary if
is the internal direct product of a
-subgroup
and a cyclic subgroup
whose order is relatively prime to
.
A group is termed elementary if it is -elementary for some prime
.
Property theory
Relation with other properties
Elementary groups are nilpotent. This is because cyclic groups are nilpotent, and -groups are also nilpotent, and a product of nilpotent groups is nilpotent.