Exercise · ZW-DIRECT-PRODUCT-OF-GROUPS-005 · ★☆☆ Basic
When is the direct product \(G\times H\) abelian?
Choose one option.
Show answer
- A. Exactly when \(G\) and \(H\) are both abelian
Explanation
Since the operation is componentwise, \((g_1,h_1)(g_2,h_2)=(g_2,h_2)(g_1,h_1)\) for all pairs is equivalent to \(g_1g_2=g_2g_1\) and \(h_1h_2=h_2h_1\). If one factor is not abelian, neither is the product.
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-006 What is the order of (3,2) in ℤ_8×ℤ_12?
- ZW-DIRECT-PRODUCT-OF-GROUPS-007 Which of the following groups is cyclic?
- 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?
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