IIT JAM Exam > IIT JAM Questions > Let G be a finite group and o(G) denotes its ...
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Let G be a finite group and o(G) denotes its order. Then which of the following statement(s) is(are) TRUE?
- a)G is abelian if o(G) where p and q are distinct primes
- b)G is abelian if every non identity element of G is of order 2
- c)G is abelian if the quotient group G/z(G)
is cyclic, where Z(G) is the center of G - d)G is abelian if o(G) = p3 , where p is prime
Correct answer is option 'B,C'. Can you explain this answer?
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Let G be a finite group and o(G) denotes its order. Then which of the ...
**Statement a) G is abelian if o(G) where p and q are distinct primes**
This statement is not true. The order of a group does not determine whether it is abelian or not. There are non-abelian groups with prime orders, such as the symmetric group S3.
**Statement b) G is abelian if every non identity element of G is of order 2**
This statement is true. If every non-identity element of G is of order 2, then G must be an abelian group. This is because in an abelian group, the product of any two elements commutes, so the order of their product is the least common multiple of their orders. Since every non-identity element has order 2, the product of any two elements will also have order 2, and thus the group is abelian.
**Statement c) G is abelian if the quotient group G/Z(G) is cyclic, where Z(G) is the center of G**
This statement is also true. The center of a group consists of the elements that commute with every element of the group. If the quotient group G/Z(G) is cyclic, then every coset of Z(G) in G is a single element. This means that every element of G commutes with every other element of G, and thus G is abelian.
**Statement d) G is abelian if o(G) = p^3, where p is prime**
This statement is not true. The order of a group does not determine whether it is abelian or not. There are non-abelian groups of order p^3, such as the Heisenberg group, which is a group of upper triangular matrices.
In summary, statements b) and c) are true. Statement a) is false, and statement d) is also false.
This statement is not true. The order of a group does not determine whether it is abelian or not. There are non-abelian groups with prime orders, such as the symmetric group S3.
**Statement b) G is abelian if every non identity element of G is of order 2**
This statement is true. If every non-identity element of G is of order 2, then G must be an abelian group. This is because in an abelian group, the product of any two elements commutes, so the order of their product is the least common multiple of their orders. Since every non-identity element has order 2, the product of any two elements will also have order 2, and thus the group is abelian.
**Statement c) G is abelian if the quotient group G/Z(G) is cyclic, where Z(G) is the center of G**
This statement is also true. The center of a group consists of the elements that commute with every element of the group. If the quotient group G/Z(G) is cyclic, then every coset of Z(G) in G is a single element. This means that every element of G commutes with every other element of G, and thus G is abelian.
**Statement d) G is abelian if o(G) = p^3, where p is prime**
This statement is not true. The order of a group does not determine whether it is abelian or not. There are non-abelian groups of order p^3, such as the Heisenberg group, which is a group of upper triangular matrices.
In summary, statements b) and c) are true. Statement a) is false, and statement d) is also false.
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Let G be a finite group and o(G) denotes its order. Then which of the following statement(s) is(are) TRUE?a)G is abelian ifo(G) where pand q are distinct primesb)G is abelian if every non identity element of G is of order 2c)G is abelian if the quotient group G/z(G)is cyclic, where Z(G)is the center of Gd)G is abelian if o(G) = p3, where p is primeCorrect answer is option 'B,C'. Can you explain this answer?
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Let G be a finite group and o(G) denotes its order. Then which of the following statement(s) is(are) TRUE?a)G is abelian ifo(G) where pand q are distinct primesb)G is abelian if every non identity element of G is of order 2c)G is abelian if the quotient group G/z(G)is cyclic, where Z(G)is the center of Gd)G is abelian if o(G) = p3, where p is primeCorrect answer is option 'B,C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let G be a finite group and o(G) denotes its order. Then which of the following statement(s) is(are) TRUE?a)G is abelian ifo(G) where pand q are distinct primesb)G is abelian if every non identity element of G is of order 2c)G is abelian if the quotient group G/z(G)is cyclic, where Z(G)is the center of Gd)G is abelian if o(G) = p3, where p is primeCorrect answer is option 'B,C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a finite group and o(G) denotes its order. Then which of the following statement(s) is(are) TRUE?a)G is abelian ifo(G) where pand q are distinct primesb)G is abelian if every non identity element of G is of order 2c)G is abelian if the quotient group G/z(G)is cyclic, where Z(G)is the center of Gd)G is abelian if o(G) = p3, where p is primeCorrect answer is option 'B,C'. Can you explain this answer?.
Let G be a finite group and o(G) denotes its order. Then which of the following statement(s) is(are) TRUE?a)G is abelian ifo(G) where pand q are distinct primesb)G is abelian if every non identity element of G is of order 2c)G is abelian if the quotient group G/z(G)is cyclic, where Z(G)is the center of Gd)G is abelian if o(G) = p3, where p is primeCorrect answer is option 'B,C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let G be a finite group and o(G) denotes its order. Then which of the following statement(s) is(are) TRUE?a)G is abelian ifo(G) where pand q are distinct primesb)G is abelian if every non identity element of G is of order 2c)G is abelian if the quotient group G/z(G)is cyclic, where Z(G)is the center of Gd)G is abelian if o(G) = p3, where p is primeCorrect answer is option 'B,C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a finite group and o(G) denotes its order. Then which of the following statement(s) is(are) TRUE?a)G is abelian ifo(G) where pand q are distinct primesb)G is abelian if every non identity element of G is of order 2c)G is abelian if the quotient group G/z(G)is cyclic, where Z(G)is the center of Gd)G is abelian if o(G) = p3, where p is primeCorrect answer is option 'B,C'. Can you explain this answer?.
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ample number of questions to practice Let G be a finite group and o(G) denotes its order. Then which of the following statement(s) is(are) TRUE?a)G is abelian ifo(G) where pand q are distinct primesb)G is abelian if every non identity element of G is of order 2c)G is abelian if the quotient group G/z(G)is cyclic, where Z(G)is the center of Gd)G is abelian if o(G) = p3, where p is primeCorrect answer is option 'B,C'. Can you explain this answer? tests, examples and also practice IIT JAM tests.
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