Exercise · ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-003 · ★☆☆ Basic
Which of these groups is isomorphic to \(\mathbb{Z}_{12}\)?
Choose one option.
Show answer
- B. \(\mathbb{Z}_{3}\times\mathbb{Z}_{4}\)
Explanation
Since \(\gcd(3,4)=1\), \(\mathbb{Z}_{3}\times\mathbb{Z}_{4}\) is cyclic of order \(12\). \(\mathbb{Z}_{2}\times\mathbb{Z}_{6}\) is not cyclic and the other two are not abelian.
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
- ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-001 How many abelian groups of order 12 are there, up to isomorphism?
- ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-002 How many abelian groups of order 16 are there, up to isomorphism?
- ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-004 Decompose ℤ_6×ℤ_10 into cyclic factors of prime-power order.
- ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-005 How many abelian groups of order 35 are there, up to isomorphism?
- ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-006 What is the largest order of an element of ℤ_4×ℤ_6?
- ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-007 How many abelian groups of order 72 are there, up to isomorphism?
- ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-008 Which pair of groups is isomorphic?
- ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-009 How many elements of order 2 does ℤ_2×ℤ_2×ℤ_4 have?
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