An elementary abelian group is a group that is both abelian and of exponent a prime number $p$; that is, every non‑identity element has order exactly $p$. Equivalently, it is a vector space over the finite field $\mathbb{F}_{p}$ with the group operation corresponding to vector addition.
Definition
Let $p$ be a prime. A group $G$ is called elementary abelian (or an elementary abelian $p$-group) if
- $G$ is abelian, and
- for every $g\in G$ with $g eq e$, the order of $g$ is $p$.
In the finite case, such a group can be written as a direct product of $n$ copies of the cyclic group of order $p$: $$ G \cong C_{p} \times C_{p} \times \cdots \times C_{p}; (n\text{ factors}) . $$ Hence $|G| = p^{,n}$ and the group has rank $n$, which equals the dimension of the associated $\mathbb{F}_{p}$-vector space.
Basic Properties
| Property | Description |
|---|---|
| Structure | $G \cong (\mathbb{Z}/p\mathbb{Z})^{n}$ for some non‑negative integer $n$. |
| Exponent | The exponent of $G$ is $p$. |
| Abelian | By definition, the group operation is commutative. |
| Subgroups | Every subgroup is itself elementary abelian; subgroups correspond to subspaces of the vector space $\mathbb{F}_{p}^{n}$. |
| Automorphisms | $\operatorname{Aut}(G) \cong \operatorname{GL}(n,\mathbb{F}{p})$, the general linear group of degree $n$ over $\mathbb{F}{p}$. |
| Classification | Finite elementary abelian groups are classified uniquely by the pair $(p,n)$. |
| Sylow Subgroups | In any finite group $H$, a Sylow $p$-subgroup that is elementary abelian is the direct product of $n$ copies of $C_{p}$. |
Examples
- $C_{p}$ – the cyclic group of prime order $p$ is elementary abelian (the case $n=1$).
- $C_{2}\times C_{2}$ – the Klein four‑group, often denoted $V_{4}$, is elementary abelian with $p=2,;n=2$.
- $C_{3}\times C_{3}\times C_{3}$ – an elementary abelian 3‑group of order $3^{3}=27$.
Relationship to Other Concepts
- Elementary abelian $p$-group – the term “elementary abelian group’’ is typically used when a specific prime $p$ is understood; the qualifier “$p$-group’’ makes the prime explicit.
- Vector spaces over finite fields – the correspondence $G \leftrightarrow \mathbb{F}_{p}^{n}$ provides a bridge between group theory and linear algebra, allowing many group‑theoretic questions to be treated with linear‑algebraic methods.
- Burnside’s $p^{a}q^{b}$ theorem – elementary abelian groups appear as minimal non‑trivial normal subgroups in the proof of certain solvability results.
- Group cohomology – the cohomology groups $H^{*}(G,\mathbb{F}_{p})$ for an elementary abelian group $G$ are polynomial algebras, a fact used in the classification of finite groups of Lie type.
Applications
Elementary abelian groups serve as building blocks in the classification of finite abelian groups, in the study of group actions (especially as regular $p$-groups), and in the analysis of automorphism groups of combinatorial structures such as designs and codes. Their linear‑algebraic nature makes them convenient in computational group theory; algorithms for testing isomorphism, enumerating subgroups, and computing automorphisms often reduce to matrix calculations over $\mathbb{F}_{p}$.
References
- D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed., Wiley, 2004.
- J. J. Rotman, An Introduction to the Theory of Groups, 4th ed., Springer, 1995.
- M. Aschbacher, Finite Group Theory, Cambridge University Press, 2000.
- J. L. Alperin and R. B. Bell, Groups and Representations, Springer, 1995.
(These sources provide formal definitions, structural theorems, and further discussion on elementary abelian groups.)