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question: If $\phi$ is a homomorphism from $\mathbb{Z}_{30}$ onto a group of order 5, determine the kernel of $\phi$.

The kernel is a normal subgroup of $\mathbb{Z}_{30}$ and this is a subgroup of $\mathbb{Z}_{30}$.

$\left | H \right |=5$ implies the element are of order possibly 1,5

Hint is appreciated. Thanks in advance.

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Let $G$ be a group of order $5$.
By Fundamental Theorem of Homomorphism, $$\mathbb{Z}_{30}/\ker \phi \cong G$$ Hence $|\ker \phi|=|\Bbb{Z}_{30}|/|G|=6$.
We conclude that $\ker \phi=\langle 5\rangle$, since it is the unique subgroup of order $6$ in $\Bbb{Z}_{30}$.

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