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Let I be the additive group of integers. Let H = {3x/x € 1} normal subgroup of I, the elements of I/H are
(a) {H,H+1,H+2} (b) {H+1, H+2, +3, H}
(c) {H, H+1,H+3} (d) {H,H+1,H+4}?
(a) {H,H+1,H+2} (b) {H+1, H+2, +3, H}
(c) {H, H+1,H+3} (d) {H,H+1,H+4}?
Most Upvoted Answer
Let I be the additive group of integers. Let H = {3x/x € 1} normal sub...
Explanation:
Identification of Elements in I/H:
- The elements in I/H are the cosets of H in I.
- The cosets are of the form a + H, where a is an integer.
Determination of Elements in I/H:
- Let's consider the elements in H: H = {3x | x ∈ Z} = {..., -6, -3, 0, 3, 6, ...}
- The cosets of H in I are:
- H: {..., -6, -3, 0, 3, 6, ...}
- H + 1: {..., -5, -2, 1, 4, 7, ...}
- H + 2: {..., -4, -1, 2, 5, 8, ...}
Identification of Correct Answer:
- Comparing the elements in I/H with the provided options:
- Option (a): {H, H + 1, H + 2} does not include all possible cosets.
- Option (b): {H + 1, H + 2, +3, H} includes extra element +3.
- Option (c): {H, H + 1, H + 3} is missing the coset H + 2.
- Option (d): {H, H + 1, H + 4} includes an extra element H + 4.
Correct Answer:
The correct answer is (b) {H + 1, H + 2, +3, H}.
Identification of Elements in I/H:
- The elements in I/H are the cosets of H in I.
- The cosets are of the form a + H, where a is an integer.
Determination of Elements in I/H:
- Let's consider the elements in H: H = {3x | x ∈ Z} = {..., -6, -3, 0, 3, 6, ...}
- The cosets of H in I are:
- H: {..., -6, -3, 0, 3, 6, ...}
- H + 1: {..., -5, -2, 1, 4, 7, ...}
- H + 2: {..., -4, -1, 2, 5, 8, ...}
Identification of Correct Answer:
- Comparing the elements in I/H with the provided options:
- Option (a): {H, H + 1, H + 2} does not include all possible cosets.
- Option (b): {H + 1, H + 2, +3, H} includes extra element +3.
- Option (c): {H, H + 1, H + 3} is missing the coset H + 2.
- Option (d): {H, H + 1, H + 4} includes an extra element H + 4.
Correct Answer:
The correct answer is (b) {H + 1, H + 2, +3, H}.
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Let I be the additive group of integers. Let H = {3x/x € 1} normal subgroup of I, the elements of I/H are(a) {H,H+1,H+2} (b) {H+1, H+2, +3, H}(c) {H, H+1,H+3} (d) {H,H+1,H+4}?
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Let I be the additive group of integers. Let H = {3x/x € 1} normal subgroup of I, the elements of I/H are(a) {H,H+1,H+2} (b) {H+1, H+2, +3, H}(c) {H, H+1,H+3} (d) {H,H+1,H+4}? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let I be the additive group of integers. Let H = {3x/x € 1} normal subgroup of I, the elements of I/H are(a) {H,H+1,H+2} (b) {H+1, H+2, +3, H}(c) {H, H+1,H+3} (d) {H,H+1,H+4}? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let I be the additive group of integers. Let H = {3x/x € 1} normal subgroup of I, the elements of I/H are(a) {H,H+1,H+2} (b) {H+1, H+2, +3, H}(c) {H, H+1,H+3} (d) {H,H+1,H+4}?.
Let I be the additive group of integers. Let H = {3x/x € 1} normal subgroup of I, the elements of I/H are(a) {H,H+1,H+2} (b) {H+1, H+2, +3, H}(c) {H, H+1,H+3} (d) {H,H+1,H+4}? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let I be the additive group of integers. Let H = {3x/x € 1} normal subgroup of I, the elements of I/H are(a) {H,H+1,H+2} (b) {H+1, H+2, +3, H}(c) {H, H+1,H+3} (d) {H,H+1,H+4}? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let I be the additive group of integers. Let H = {3x/x € 1} normal subgroup of I, the elements of I/H are(a) {H,H+1,H+2} (b) {H+1, H+2, +3, H}(c) {H, H+1,H+3} (d) {H,H+1,H+4}?.
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