Elementary abelian group
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This article defines a group property: a property that can be evaluated to true/false for any given group
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Contents |
Definition
Symbol-free definition
An elementary Abelian group is a group that satisfies the following equivalent conditions:
- It is an Abelian characteristically simple group
- It is a direct product of isomorphic subgroups, each being cyclic of prime order
- It is the additive group of a vector space over a prime field
Relation with other properties
Stronger properties
- Cyclic group of prime order viz. simple Abelian group
Weaker properties
Facts
Minimal normal subgroups
Any minimal normal subgroup in a solvable group must be elementary Abelian. This follows by combining the fact that it must be Abelian with the fact that in any group, a minimal normal subgroup is always characteristically simple.
