The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or false?
The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or false?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Question:**
The group generated by the cycle \( (1, 2) \) is a normal subgroup of the symmetric group \( S_3 \). True or false?
- [ ] True
- [ ] False
**Explanation of Question:**
This question tests understanding of group theory concepts, specifically the properties of subgroups within symmetric groups. The symmetric group \( S_3 \) consists of all permutations of three elements. A subgroup is called normal if it is invariant under conjugation by any element of the group. The cycle \( (1, 2) \) refers to the permutation that swaps elements 1 and 2, leaving all others fixed. The question asks whether the group generated by \( (1, 2) \) is a normal subgroup of \( S_3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde776b38-607a-41e9-8f7e-215921048344%2F1f3fff4f-b2b8-499f-adb2-785ac1bca723%2Fd94nxg6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
The group generated by the cycle \( (1, 2) \) is a normal subgroup of the symmetric group \( S_3 \). True or false?
- [ ] True
- [ ] False
**Explanation of Question:**
This question tests understanding of group theory concepts, specifically the properties of subgroups within symmetric groups. The symmetric group \( S_3 \) consists of all permutations of three elements. A subgroup is called normal if it is invariant under conjugation by any element of the group. The cycle \( (1, 2) \) refers to the permutation that swaps elements 1 and 2, leaving all others fixed. The question asks whether the group generated by \( (1, 2) \) is a normal subgroup of \( S_3 \).
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