Abstract Algebra

JKPSC Assistant Professor (Mathematics) · Mathematics · 64 practice questions with answers and explanations: 34 Easy, 30 Medium, 0 Hard.

Sample questions

Sample Easy question 1

A subgroup H of a finite group G has order 4, and |G| = 36. What is the index [G : H]?

  1. 9
  2. 4
  3. 10
  4. 32
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Answer: A. [G : H] = |G|/|H| = 36/4 = 9.

Sample Easy question 2

What is the order of the element 8 in the additive group Z30?

  1. 30
  2. 8
  3. 15
  4. 16
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Answer: C. The order of k in Z_n is n/gcd(n, k) = 30/2 = 15.

Sample Easy question 3

How many generators does the cyclic group of order 15 have?

  1. 7
  2. 14
  3. 8
  4. 10
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Answer: C. The number of generators is Euler's φ(15) = 8.

Sample Medium question 4

By Lagrange's theorem, a group of order 30 CANNOT have a subgroup of order:

  1. 15
  2. 7
  3. 3
  4. 2
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Answer: B. The order of a subgroup must divide 30; 7 does not.

Sample Medium question 5

How many subgroups does the cyclic group of order 12 have?

  1. 12
  2. 7
  3. 6
  4. 5
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Answer: C. A cyclic group of order n has one subgroup for each divisor of n; 12 has 6 divisors.

Sample Medium question 6

How many non-zero zero divisors does the ring Z15 have?

  1. 7
  2. 6
  3. 5
  4. 14
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Answer: B. A non-zero element k is a zero divisor exactly when gcd(k, 15) > 1; there are 6.

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