Mathematics Exam > Mathematics Questions > Let G be a group of order 143,then the centre...
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Let G be a group of order 143, then the centre of G is isomorphic to
- a)Z
- b)Z11
- c)Z13
- d)Z143
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Let G be a group of order 143,then the centre of G is isomorphic toa)Z...
O(G) = 143
i.e. O(G) =11 x 13 (p < q)
Here.O(G) = pq , p < q and p X q -1, so G be a cyclic group of order 143 and every cyclic group is an abelian group.
Hence G is an abelian group, if G be an abelian group then centre of G is equal to group G.
G = Z(G) [∵ G is an abelian]
i.e. O[Z(G)] = 143 so Z(G) is isomorphic to Z143.
i.e. O(G) =11 x 13 (p < q)
Here.O(G) = pq , p < q and p X q -1, so G be a cyclic group of order 143 and every cyclic group is an abelian group.
Hence G is an abelian group, if G be an abelian group then centre of G is equal to group G.
G = Z(G) [∵ G is an abelian]
i.e. O[Z(G)] = 143 so Z(G) is isomorphic to Z143.
Most Upvoted Answer
Let G be a group of order 143,then the centre of G is isomorphic toa)Z...
Explanation:
There are some key points to consider when determining the isomorphism of the center of a group of order 143:
- Order of the Group:
- The order of the group G is 143, which is a prime number.
- Any group of prime order is always cyclic.
- Center of a Group:
- The center of a group G, denoted by Z(G), is the set of all elements that commute with every element of the group.
- In a cyclic group, every element commutes with every other element in the group. Therefore, the center of a cyclic group is the group itself.
- Isomorphism:
- Two groups are isomorphic if there exists a bijective group homomorphism between them.
- Since the center of a cyclic group is the group itself, it is isomorphic to the group itself.
- Conclusion:
- Since the group G of order 143 is cyclic, its center is isomorphic to the group itself.
- Therefore, the center of G is isomorphic to G itself, which is the cyclic group of order 143 (Z143).
Therefore, the correct answer is option D) Z143.
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Community Answer
Let G be a group of order 143,then the centre of G is isomorphic toa)Z...
O(G) = 143
i.e. O(G) =11 x 13 (p < q)
Here.O(G) = pq , p < q and p X q -1, so G be a cyclic group of order 143 and every cyclic group is an abelian group.
Hence G is an abelian group, if G be an abelian group then centre of G is equal to group G.
G = Z(G) [∵ G is an abelian]
i.e. O[Z(G)] = 143 so Z(G) is isomorphic to Z143.
i.e. O(G) =11 x 13 (p < q)
Here.O(G) = pq , p < q and p X q -1, so G be a cyclic group of order 143 and every cyclic group is an abelian group.
Hence G is an abelian group, if G be an abelian group then centre of G is equal to group G.
G = Z(G) [∵ G is an abelian]
i.e. O[Z(G)] = 143 so Z(G) is isomorphic to Z143.
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Let G be a group of order 143,then the centre of G is isomorphic toa)Zb)Z11c)Z13d)Z143Correct answer is option 'D'. Can you explain this answer?
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Let G be a group of order 143,then the centre of G is isomorphic toa)Zb)Z11c)Z13d)Z143Correct answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let G be a group of order 143,then the centre of G is isomorphic toa)Zb)Z11c)Z13d)Z143Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a group of order 143,then the centre of G is isomorphic toa)Zb)Z11c)Z13d)Z143Correct answer is option 'D'. Can you explain this answer?.
Let G be a group of order 143,then the centre of G is isomorphic toa)Zb)Z11c)Z13d)Z143Correct answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let G be a group of order 143,then the centre of G is isomorphic toa)Zb)Z11c)Z13d)Z143Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a group of order 143,then the centre of G is isomorphic toa)Zb)Z11c)Z13d)Z143Correct answer is option 'D'. Can you explain this answer?.
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