The Group Properties Wiki (pre-alpha)

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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 1 Definition 1.1 Symbol-free definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Facts 3.1 Minimal normal subgroups

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

• Homocyclic group
• Abelian group
• Characteristically simple group

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.