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Groups and Subgroups

This quiz covers fundamental concepts and properties related to groups and subgroups in abstract algebra.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Let G be a group and H a subset of G. Which of the following conditions is necessary for H to be a subgroup of G?

  1. H is non-empty.
  2. H is closed under the group operation.
  3. H contains the identity element of G.
  4. All of the above.
Question 2 Multiple Choice (Single Answer)

If H is a subgroup of a group G, then the order of H (|H|) is:

  1. Always a factor of the order of G (|G|).
  2. Always less than or equal to the order of G (|G|).
  3. Always greater than or equal to the order of G (|G|).
  4. None of the above.
Question 3 Multiple Choice (Single Answer)

Which of the following is NOT a necessary condition for a subset H of a group G to be a subgroup?

  1. H is non-empty.
  2. H is closed under the group operation.
  3. H contains the identity element of G.
  4. H contains the inverse of every element in H.
Question 4 Multiple Choice (Single Answer)

The intersection of two subgroups of a group G is:

  1. Always a subgroup of G.
  2. Sometimes a subgroup of G.
  3. Never a subgroup of G.
  4. None of the above.
Question 5 Multiple Choice (Single Answer)

The union of two subgroups of a group G is:

  1. Always a subgroup of G.
  2. Sometimes a subgroup of G.
  3. Never a subgroup of G.
  4. None of the above.
Question 6 Multiple Choice (Single Answer)

Let G be a group and H a subgroup of G. The set of all left cosets of H in G is denoted by:

  1. G/H
  2. H/G
  3. G\H
  4. H\G
Question 7 Multiple Choice (Single Answer)

Let G be a group and H a subgroup of G. The set of all right cosets of H in G is denoted by:

  1. G/H
  2. H/G
  3. G\H
  4. H\G
Question 8 Multiple Choice (Single Answer)

If H is a subgroup of a group G, then the index of H in G (|G:H|) is:

  1. The number of elements in H.
  2. The number of elements in G.
  3. The number of cosets of H in G.
  4. None of the above.
Question 9 Multiple Choice (Single Answer)

Let G be a group and H a subgroup of G. The normalizer of H in G, denoted by $N_G(H)$, is:

  1. The set of all elements in G that commute with every element in H.
  2. The set of all elements in G that conjugate H.
  3. The set of all elements in G that are contained in some conjugate of H.
  4. All of the above.
Question 10 Multiple Choice (Single Answer)

Let G be a group and H a subgroup of G. The centralizer of H in G, denoted by $C_G(H)$, is:

  1. The set of all elements in G that commute with every element in H.
  2. The set of all elements in G that conjugate H.
  3. The set of all elements in G that are contained in some conjugate of H.
  4. None of the above.
Question 11 Multiple Choice (Single Answer)

A group G is called abelian if:

  1. Every element in G has order 2.
  2. Every element in G commutes with every other element in G.
  3. Every subgroup of G is normal.
  4. None of the above.
Question 12 Multiple Choice (Single Answer)

A group G is called cyclic if:

  1. It is generated by a single element.
  2. It is abelian.
  3. It is finite.
  4. None of the above.
Question 13 Multiple Choice (Single Answer)

The order of an element a in a group G is:

  1. The smallest positive integer n such that $a^n = e$, where e is the identity element of G.
  2. The largest positive integer n such that $a^n = e$, where e is the identity element of G.
  3. The number of elements in the cyclic subgroup generated by a.
  4. None of the above.
Question 14 Multiple Choice (Single Answer)

Let G be a group and H a subgroup of G. If the order of H is p, where p is a prime number, then:

  1. H is a normal subgroup of G.
  2. H is a cyclic subgroup of G.
  3. Every element in H has order p.
  4. All of the above.
Question 15 Multiple Choice (Single Answer)

The Sylow theorems provide information about:

  1. The existence and number of subgroups of a given order in a finite group.
  2. The structure of finite groups.
  3. The solvability of groups.
  4. None of the above.
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