Exercise · ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-004 · ★☆☆ Basic

Decompose \(\mathbb{Z}_{6}\times\mathbb{Z}_{10}\) into cyclic factors of prime-power order.

Choose one option.

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  • A. \(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{3}\times\mathbb{Z}_{5}\)

Explanation

Split each factor: \(\mathbb{Z}_6\cong\mathbb{Z}_2\times\mathbb{Z}_3\) and \(\mathbb{Z}_{10}\cong\mathbb{Z}_2\times\mathbb{Z}_5\). The total order is \(60\), and the factor \(\mathbb{Z}_2\) appears twice.

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Review the concept: Classification of finite abelian groups

Every finite abelian group \(G\) is isomorphic to a direct product of cyclic groups \(\mathbb{Z}_{p^k}\), with \(p\) prime and \(k\ge1\), unique up to order.

\[G\cong\mathbb{Z}_{p_1^{k_1}}\times\cdots\times\mathbb{Z}_{p_r^{k_r}}\]

More practice problems: classification of finite abelian groups

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