Exercise · ZW-CLASSIFICATION-OF-FINITE-ABELIAN-GROUPS-002 · ★☆☆ Basic
How many abelian groups of order \(16\) are there, up to isomorphism?
Choose one option.
Show answer
- C. 5
Explanation
They correspond to the partitions of the exponent \(4\): \(4,\ 3+1,\ 2+2,\ 2+1+1,\ 1+1+1+1\). There are 5.
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-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?
- 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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