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If G is a finite group of order n, a belongs to G and order of a is m, if G is cyclic, then?
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If G is a finite group of order n, a belongs to G and order of a is m,...
Explanation:
Cyclic Group:
- A group G is cyclic if there exists an element a in G such that every element of G can be expressed as a power of a.
- In other words, G = {a, a^2, a^3, ..., a^n-1} where n is the order of the group.
Order of an Element:
- The order of an element a in a group G is the smallest positive integer m such that a^m = e, where e is the identity element of the group.
- If no such integer exists, then the order of the element is infinite.
Given Information:
- Let G be a finite group of order n with an element a of order m.
- We are also given that G is cyclic.
Implication:
- Since G is cyclic, there exists an element a in G such that G = {a, a^2, ..., a^n-1}.
- The order of a is m, which means a^m = e.
- Since G is cyclic, a^m = e implies that the order of G is a multiple of m.
- Therefore, the order of G, which is n, must be a multiple of m.
Conclusion:
- In a cyclic group, if an element has order m, then the order of the group must be a multiple of m.
Cyclic Group:
- A group G is cyclic if there exists an element a in G such that every element of G can be expressed as a power of a.
- In other words, G = {a, a^2, a^3, ..., a^n-1} where n is the order of the group.
Order of an Element:
- The order of an element a in a group G is the smallest positive integer m such that a^m = e, where e is the identity element of the group.
- If no such integer exists, then the order of the element is infinite.
Given Information:
- Let G be a finite group of order n with an element a of order m.
- We are also given that G is cyclic.
Implication:
- Since G is cyclic, there exists an element a in G such that G = {a, a^2, ..., a^n-1}.
- The order of a is m, which means a^m = e.
- Since G is cyclic, a^m = e implies that the order of G is a multiple of m.
- Therefore, the order of G, which is n, must be a multiple of m.
Conclusion:
- In a cyclic group, if an element has order m, then the order of the group must be a multiple of m.
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If G is a finite group of order n, a belongs to G and order of a is m, if G is cyclic, then?
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If G is a finite group of order n, a belongs to G and order of a is m, if G is cyclic, then? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If G is a finite group of order n, a belongs to G and order of a is m, if G is cyclic, then? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If G is a finite group of order n, a belongs to G and order of a is m, if G is cyclic, then?.
If G is a finite group of order n, a belongs to G and order of a is m, if G is cyclic, then? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If G is a finite group of order n, a belongs to G and order of a is m, if G is cyclic, then? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If G is a finite group of order n, a belongs to G and order of a is m, if G is cyclic, then?.
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