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A group (G, *) is called a cyclic group if there exists an element a∈G, such that every element of ‘G’ can be written an for some integer n. Then ‘a’ is called a generating element / generator. Consider the following group G = ( {1, 2, 3, 4, 5, 6},⊗7), , where ⊗7is multiplication modulo 7 operation. Let X be the number of generators of G. And Y be the sum of these generators. Find X + Y is _________________?
- a)8
- b)6
- c)5
- d)10
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A group (G, *) is called a cyclic group if there exists an element a&i...
G = ( {1, 2, 3, 4, 5, 6}, ⊗7)
The number of generators of a group (if there exist any) is Sn. Where Sn is the set of all +ve integers which are less than n and relatively prime to n. Also, n is the number of elements in G.
S6 = {1, 5} So 2 generators are there for G.
X = 2.
Y = 3 + 5 = 8, because 3 and 5 are the generators of G.
X + Y = 10.
Most Upvoted Answer
A group (G, *) is called a cyclic group if there exists an element a&i...
Such that every element in G can be expressed as a power of a. In other words, for every element g in G, there exists an integer n such that g = a^n.
Additionally, if G is finite, the order of a is equal to the order of G (the number of elements in G). If G is infinite, then the order of a is infinite.
In a cyclic group, the operation * is typically defined as multiplication. However, it can also be defined as addition, depending on the context.
Cyclic groups have various properties. For example, every subgroup of a cyclic group is cyclic. Moreover, if G is cyclic with order n, then for any positive integer k, the element a^k generates a subgroup of G of order n/gcd(n,k), where gcd denotes the greatest common divisor.
Cyclic groups are widely studied in mathematics, particularly in the field of group theory. They have numerous applications, such as in cryptography and number theory.
Additionally, if G is finite, the order of a is equal to the order of G (the number of elements in G). If G is infinite, then the order of a is infinite.
In a cyclic group, the operation * is typically defined as multiplication. However, it can also be defined as addition, depending on the context.
Cyclic groups have various properties. For example, every subgroup of a cyclic group is cyclic. Moreover, if G is cyclic with order n, then for any positive integer k, the element a^k generates a subgroup of G of order n/gcd(n,k), where gcd denotes the greatest common divisor.
Cyclic groups are widely studied in mathematics, particularly in the field of group theory. They have numerous applications, such as in cryptography and number theory.
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A group (G, *) is called a cyclic group if there exists an element a∈G, such that every element of ‘G’ can be written an for some integer n. Then ‘a’ is called a generating element / generator. Consider the following group G = ( {1, 2, 3, 4, 5, 6},⊗7), , where ⊗7is multiplication modulo 7 operation. Let X be the number of generators of G. And Y be the sum of these generators. Find X + Y is _________________?a)8b)6c)5d)10Correct answer is option 'D'. Can you explain this answer?
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A group (G, *) is called a cyclic group if there exists an element a∈G, such that every element of ‘G’ can be written an for some integer n. Then ‘a’ is called a generating element / generator. Consider the following group G = ( {1, 2, 3, 4, 5, 6},⊗7), , where ⊗7is multiplication modulo 7 operation. Let X be the number of generators of G. And Y be the sum of these generators. Find X + Y is _________________?a)8b)6c)5d)10Correct answer is option 'D'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A group (G, *) is called a cyclic group if there exists an element a∈G, such that every element of ‘G’ can be written an for some integer n. Then ‘a’ is called a generating element / generator. Consider the following group G = ( {1, 2, 3, 4, 5, 6},⊗7), , where ⊗7is multiplication modulo 7 operation. Let X be the number of generators of G. And Y be the sum of these generators. Find X + Y is _________________?a)8b)6c)5d)10Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A group (G, *) is called a cyclic group if there exists an element a∈G, such that every element of ‘G’ can be written an for some integer n. Then ‘a’ is called a generating element / generator. Consider the following group G = ( {1, 2, 3, 4, 5, 6},⊗7), , where ⊗7is multiplication modulo 7 operation. Let X be the number of generators of G. And Y be the sum of these generators. Find X + Y is _________________?a)8b)6c)5d)10Correct answer is option 'D'. Can you explain this answer?.
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