Is $\mathbb Z_7$ an extension of $\mathbb Z_2$?

Question

Is $\mathbb Z_7$ a extension of $\mathbb Z_2?$

I know it is not, because we can't establish a monomorphism between $\mathbb Z_7$ and $\mathbb Z_2$,i.e. it doesn't exist an injective homomorphism between them.

But how can I prove it formally?

Answer 1

Take a map $f : \mathbf{Z}_2 \to \mathbf{Z}_7$. If it is a homomorphism, we must have $f(0) = 0$. That just leaves open the question of what $f(1)$ is. Since it is a homomorphism we want $f(1 + 1) = f(1) + f(1)$. Is there a value of $f(1)$ for which this works?

Answer 2

Hint: Can a group of order $2$ be a subgroup of a group of order $7$ ?

If $E$ is an extension field of $F$, then the additive group of $F$ is a subgroup of the additive group of $E$.