A) (i), (ii) B C D E All (i), (iii) (ii), (iii) All Consider the subgroup 10Z of 2Z, which of the following are correct? (i): The cosets of 10Z in 2Z are 10Z,2+ 10Z,4+ 10Z,6 + 10Z,8 + 10Z. (ii): [2Z: 10Z) = 5. (iii): 10Z is a normal subgroup of 2Z.
A) (i), (ii) B C D E All (i), (iii) (ii), (iii) All Consider the subgroup 10Z of 2Z, which of the following are correct? (i): The cosets of 10Z in 2Z are 10Z,2+ 10Z,4+ 10Z,6 + 10Z,8 + 10Z. (ii): [2Z: 10Z) = 5. (iii): 10Z is a normal subgroup of 2Z.
Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Question: Consider the subgroup \(10\mathbb{Z}\) of \(2\mathbb{Z}\). Which of the following are correct?**
(i): The cosets of \(10\mathbb{Z}\) in \(2\mathbb{Z}\) are \(10\mathbb{Z}, 2 + 10\mathbb{Z}, 4 + 10\mathbb{Z}, 6 + 10\mathbb{Z}, 8 + 10\mathbb{Z}\).
(ii): \([2\mathbb{Z} : 10\mathbb{Z}] = 5\).
(iii): \(10\mathbb{Z}\) is a normal subgroup of \(2\mathbb{Z}\).
**Options:**
- A: (i), (ii)
- B: All
- C: (i), (iii)
- D: (ii), (iii)
- E: All
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Transcribed Image Text:**Question: Consider the subgroup \(10\mathbb{Z}\) of \(2\mathbb{Z}\). Which of the following are correct?**
(i): The cosets of \(10\mathbb{Z}\) in \(2\mathbb{Z}\) are \(10\mathbb{Z}, 2 + 10\mathbb{Z}, 4 + 10\mathbb{Z}, 6 + 10\mathbb{Z}, 8 + 10\mathbb{Z}\).
(ii): \([2\mathbb{Z} : 10\mathbb{Z}] = 5\).
(iii): \(10\mathbb{Z}\) is a normal subgroup of \(2\mathbb{Z}\).
**Options:**
- A: (i), (ii)
- B: All
- C: (i), (iii)
- D: (ii), (iii)
- E: All
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Expert Solution
Step 1
Elements of the group 2Z are {. . . . . ,-4, -2, 0, 2, 4, 6, 8, 10, . . . . }
So, cosets of 2Z/10Z are of the form a + 10Z, where a lies in 2Z.
Now, whenever a = 0 (mod 10),
a + 10Z = 10Z.
Whenever a = 2 (mod 10), i.e., for a = -18, -8, 2, 12,. . . . .,
a + 10Z = 2 + 10Z.
Whenever a = 4 (mod 10), i.e., for a = -16, -6, 4, 14,. . . . .,
a + 10Z = 4 + 10Z.
Whenever a = 6 (mod 10), i.e., for a = -14, -4, 6, 16,. . . . .,
a + 10Z = 6 + 10Z.
Whenever a = 8 (mod 10), i.e., for a = -12, -2, 8, 18,. . . . .,
a + 10Z = 8 + 10Z.
So, only cosets of 2Z/10Z are
10Z, 2 + 10Z, 4 + 10Z, 6 + 10Z, 8 + 10Z.
So, statement (i) is correct.
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