An elementary abelian -group is a finite abelian group in which every nonidentity element has order . It is equivalently a finite-dimensional vector space over under addition.
A group of exponent two is abelian. Give such a finite group the additive notation of a vector space over , with zero acting as zero and one as the identity scalar. The group axioms and verify the vector-space laws. A basis with vectors then gives exactly elements. Thus a finite group whose elements all have order at most two must have power-of-two order.
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An **elementary abelian group** is a specific type of group that is both abelian (commutative) and has a particular structure in which every non-identity element has an order of 2. This means that for every element \( g \) in the group, if \( g \neq e \) (where \( e \) is the identity element of the group), then \( g^2 = e \).