Exercise · ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-010 · ★★★ Advanced
Let \(G\) be an abelian group of order \(36\). Which statement is always true?
Choose one option.
Show answer
- B. \(G\) has an element of order \(6\)
Explanation
Each candidate (with a part of order \(4\) and a part of order \(9\)) has an element of order \(2\) and another of order \(3\), whose product has order \(6\). By contrast, \(\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3\times\mathbb{Z}_3\) is not cyclic and has no element of order \(4\).
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-003 Which of these groups is isomorphic to ℤ_12?
- 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?