Exercise · ZW-GENERATED-SUBGROUP-009 · ★★★ Advanced
In \((\mathbb{Z}_{18},+)\), which of the following elements generates the same subgroup as \(4\)?
Choose one option.
Show answer
- D. 8
Explanation
\(\langle 4\rangle\) is the 9 even residues of \(\mathbb{Z}_{18}\), since \(\gcd(4,18)=2\). An element generates the same subgroup when its gcd with 18 is also 2: that holds for 8. Those with gcd 6 or 9 generate smaller subgroups.
Review the concept: Generated subgroup
The subgroup generated by a subset \(S\) of a group \(G\) is the smallest subgroup \(H\) of \(G\) containing \(S\); it is denoted \(\langle S\rangle\) and \(S\) is called a generating set.
\[\langle S\rangle=\bigcap_{S\subseteq H\le G}H\]
More practice problems: generated subgroup
- ZW-GENERATED-SUBGROUP-001 In the group (ℤ,+), what is the subgroup generated by 4?
- ZW-GENERATED-SUBGROUP-002 How many elements does langle 8rangle have in (ℤ_12,+)?
- ZW-GENERATED-SUBGROUP-003 How many elements does langle 4,6rangle have in (ℤ_12,+)?
- ZW-GENERATED-SUBGROUP-004 In (ℤ,+), what is the subgroup langle 6,15rangle?
- ZW-GENERATED-SUBGROUP-005 How many elements does the subgroup of S_3 generated by the 3-cycle (1\ 2\ 3) have?
- ZW-GENERATED-SUBGROUP-006 How many elements does langle (1\ 2),(2\ 3)rangle have in S_3?
- ZW-GENERATED-SUBGROUP-007 How many elements does langle (1\ 2),(3\ 4)rangle have in S_4?
- ZW-GENERATED-SUBGROUP-008 What is the subgroup langle∅rangle generated by the empty set in a group G?