Prime-order group powers: vector spaces

powers of groups of prime order are vector spaces

Yes, that's correct — groups of prime order have an important connection to vector spaces, under the right interpretation.

Key Idea:

If $G$ is a group of prime order $p$, then:

• $G$ is a cyclic group (since all groups of prime order are cyclic),
• Every element $g \in G$ satisfies $g^p = e$, the identity element,
• If $G$ is also abelian and we consider it as a module over $\mathbb{F}_p$, then it becomes a vector space over $\mathbb{F}_p$.

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Explanation:

Let’s formalize it:

1. Group of Prime Order:

Let $G$ be a group with order $p$, a prime number.

• Then $G$ is cyclic: $G \cong \mathbb{Z}/p\mathbb{Z}$,
• It is abelian, because all cyclic groups are abelian.

2. Powers and Vector Space Structure:

If you now look at a finite abelian group $G$ where every element has order dividing a prime $p$, i.e., for all $g \in G$, $g^p = e$, then:

• $G$ is a $\mathbb{F}_p$-vector space.

Why?

Because we can define scalar multiplication by elements of $\mathbb{F}_p$ as:

$$ a \cdot g = g^a $$

This satisfies the vector space axioms because exponentiation respects the group operation.

Thus:

Any abelian group $G$ where every element has order $p$ (a prime) can be regarded as a vector space over $\mathbb{F}_p$.

This includes:

• Any group of order $p$,
• More generally, any abelian $p$-group of exponent $p$.

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Examples:

• $\mathbb{Z}/p\mathbb{Z} \cong \mathbb{F}_p$: 1-dimensional vector space over $\mathbb{F}_p$,
• $(\mathbb{Z}/p\mathbb{Z})^n \cong \mathbb{F}_p^n$: n-dimensional vector space over $\mathbb{F}_p$.

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Conclusion:

So yes — the powers (i.e., elements raised to integers) in abelian groups of prime order, or more generally abelian $p$-groups of exponent $p$, behave like elements of a vector space over the finite field $\mathbb{F}_p$.