Exercise · ZW-SUBGROUP-009 · ★★☆ Intermediate
Which is a subgroup of \((\mathbb{Z}_7^*,\cdot)\), the integers \(1,\dots,6\) under multiplication mod 7?
Choose one option.
Show answer
- A. \(\{1,2,4\}\)
Explanation
In \(\{1,2,4\}\): \(2\cdot2=4\), \(2\cdot4=1\), \(4\cdot4=2\), and every element has its inverse inside. In \(\{1,3\}\) the product \(3\cdot3=2\) leaves, and the other two fail by \(2\cdot3=6\) and \(2\cdot5=3\).
Review the concept: Subgroup
A subset \(H\) of a group \(G\) that is itself a group under the operation of \(G\).
\[H\le G\]
More practice problems: subgroup
- ZW-SUBGROUP-001 Which of the following sets is a subgroup of (ℤ,+)?
- ZW-SUBGROUP-002 What condition is enough for a subset H of a group G to be a subgroup?
- ZW-SUBGROUP-003 Which of the following is NOT a subgroup of (ℤ_12,+)?
- ZW-SUBGROUP-004 How many subgroups does (ℤ_12,+) have, counting the trivial one and the whole group?
- ZW-SUBGROUP-005 Which is a subgroup of (ℝ^*,·), the nonzero reals under multiplication?
- ZW-SUBGROUP-006 Let H=2ℤ and K=3ℤ in (ℤ,+). Which statement is true?
- ZW-SUBGROUP-007 The smallest subgroup of (ℤ,+) containing 4 and 6 is
- ZW-SUBGROUP-008 Which of these subsets of GL_2(ℝ) is a subgroup?