Exercise · ZW-DIRECT-PRODUCT-OF-GROUPS-007 · ★★☆ Intermediate
Which of the following groups is cyclic?
Choose one option.
Show answer
- B. \(\mathbb{Z}_3\times\mathbb{Z}_8\)
Explanation
An element \((a,b)\) generates all of \(\mathbb{Z}_m\times\mathbb{Z}_n\) only if its order \(\operatorname{lcm}\) reaches \(mn\), which requires \(\gcd(m,n)=1\). Here \(\gcd(3,8)=1\), whereas the other three pairs of moduli share a factor 2 or 3.
Review the concept: Direct product of groups
The direct product \(G\times H\) of two groups \(G\) and \(H\) is the set of pairs \((g,h)\), with \(g\in G\) and \(h\in H\), with the operation defined componentwise.
\[(g_1,h_1)(g_2,h_2)=(g_1g_2,\;h_1h_2)\]
More practice problems: direct product of groups
- ZW-DIRECT-PRODUCT-OF-GROUPS-001 How many elements does ℤ_4×ℤ_6 have?
- ZW-DIRECT-PRODUCT-OF-GROUPS-002 What is the identity element of ℤ_5×ℤ_7 under componentwise addition?
- ZW-DIRECT-PRODUCT-OF-GROUPS-003 In ℤ_6×ℤ_4, compute (2,3)+(5,2).
- ZW-DIRECT-PRODUCT-OF-GROUPS-004 What is the inverse of (3,2) in ℤ_7×ℤ_5?
- ZW-DIRECT-PRODUCT-OF-GROUPS-005 When is the direct product G× H abelian?
- ZW-DIRECT-PRODUCT-OF-GROUPS-006 What is the order of (3,2) in ℤ_8×ℤ_12?
- ZW-DIRECT-PRODUCT-OF-GROUPS-008 How many elements of order 2 does ℤ_2×ℤ_2 have?
- ZW-DIRECT-PRODUCT-OF-GROUPS-009 How many elements of order 3 are there in ℤ_3×ℤ_3?
← ZW-DIRECT-PRODUCT-OF-GROUPS-006 ZW-DIRECT-PRODUCT-OF-GROUPS-008 →