GATE Exam > GATE Questions > A group (G, *) is called a cyclic group if th...
Start Learning for Free
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?
| FREE This question is part of | Download PDF Attempt this Test |
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.
|
Explore Courses for GATE exam
|
|
Similar GATE Doubts
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?
Question Description
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?.
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?.
Solutions 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? in English & in Hindi are available as part of our courses for GATE.
Download more important topics, notes, lectures and mock test series for GATE Exam by signing up for free.
Here you can find the meaning of 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? defined & explained in the simplest way possible. Besides giving the explanation of
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?, a detailed solution 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? has been provided alongside types of 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? theory, EduRev gives you an
ample number of questions to practice 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? tests, examples and also practice GATE tests.
|
|
Explore Courses for GATE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup