Elementary abelian group

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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 restricted direct product of isomorphic subgroups, each being cyclic of prime order
• It is the additive group of a vector space over a prime field
• It is abelian and all elements other than the identity have the same order

Examples

Finite groups

The finite elementary abelian groups are precisely those of the form $(\mathbb {Z} _{p})^{n}$ for some $p,n\in \mathbb {Z}$, $p$ prime.

• In particular, all cyclic groups of prime order are elementary abelian groups. Since there is only one group up to isomorphism of prime order, which is cyclic, all groups of prime order are elementary abelian groups.

• Elementary abelian group:E4, the direct product $\mathbb {Z} _{2}\times \mathbb {Z} _{2}$, also known as the Klein four-group.
• Elementary abelian group:E8, the direct product $\mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}$
• Elementary abelian group:E9, the direct product $\mathbb {Z} _{3}\times \mathbb {Z} _{3}$
• Elementary abelian group:E16, the direct product $\mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}$
• Elementary abelian group:E25, the direct product $\mathbb {Z} _{5}\times \mathbb {Z} _{5}$

Infinite groups

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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.