Exercise · ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-009 · ★★★ Advanced
How many elements of order \(2\) does \(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}\) have?
Choose one option.
Show answer
- C. 7
Explanation
The elements with \(2x=0\) form \(\mathbb{Z}_2\times\mathbb{Z}_2\times\{0,2\}\), with \(8\) elements. Removing the identity leaves \(7\).
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?
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- 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.
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- 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?
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