GATE MA topic:algebra PYQs & Answers — Page 3

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Questions 21–30 of 54. Imported answers require independent review unless marked reviewed.

Topic practice 54 questionsPage 3 of 6

Let

be an abelian group and

be a surjective group homomorphism. Let

for some

. Consider the following statements:

: For every

, there exists an

such that

: Let

be the identity of

and

be the subgroup generated by

. Then

and

Which of the following statements is/are correct?

Your answerSelect all correct options. You can choose more than one.

is TRUE

is FALSE

is TRUE

is FALSE

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Let

, where

is the permutation group on 8 elements. Suppose

is the product of

and

, where

is a 4-cycle and

is a 3-cycle in

. If

and

are disjoint cycles, then the number of elements in

which are conjugate to

is __________.

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The number of non-isomorphic abelian groups of order

is ________.

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The number of subgroups of a cyclic group of order

is ________.

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Let

denote the ring of polynomials in

with real coefficients. Then, the quotient ring

Your answerChoose one option.

a field

an integral domain, but not a field

not an integral domain, but has 0 as the only nilpotent element

a ring which contains non-zero nilpotent elements

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Consider the following conditions on two proper non-zero ideals

and

of a non-zero commutative ring

.\n\nP: For any

, there exists a unique

such that

and

.\n\nQ:

\n\nThen, which of the following statements is TRUE?

Your answerChoose one option.

P implies Q but Q does not imply P

Q implies P but P does not imply Q

P implies Q and Q implies P

P does not imply Q and Q does not imply P

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Let

denote the subset of

consisting of elements algebraic over

. Then, which of the following statements are TRUE?

Your answerSelect all correct options. You can choose more than one.

No element of

is algebraic over

is an algebraically closed field

For any bijective ring homomorphism

, we have

There is no bijection between

and

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The number of 5-Sylow subgroups in the symmetric group

of degree 5 is _____.

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Let

be the ideal generated by

in the polynomial ring

, where

denotes the ring of integers modulo 3. Then the number of units in the quotient ring

is ______.

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Let

denote the ring of integers. Consider the subring

of the field

of complex numbers. Consider the following statements: P:

is an irreducible element. Q:

is a prime element. Then

Your answerChoose one option.

both P and Q are TRUE

P is TRUE and Q is FALSE

P is FALSE and Q is TRUE

both P and Q are FALSE

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GATE MA topic:algebra PYQs

Question 54

Question 58

Question 26

Question 27

Question 36

Question 37

Question 49

Question 17

Question 18

Question 26