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Groups and Subgroups

Description: This quiz covers fundamental concepts and properties related to groups and subgroups in abstract algebra.
Number of Questions: 15
Created by:
Tags: groups subgroups group theory abstract algebra
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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.

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Correct Option: D
Explanation:

For H to be a subgroup of G, it must satisfy all three conditions: non-emptiness, closure under the group operation, and containing the identity element.

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.

Show answer
Correct Option: A
Explanation:

Lagrange's Theorem states that the order of a subgroup H of a group G divides the order of G.

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.

Show answer
Correct Option: D
Explanation:

The condition of containing the inverse of every element in H is not necessary for H to be a subgroup. However, it is a necessary condition for H to be a normal subgroup.

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.

Show answer
Correct Option: A
Explanation:

The intersection of two subgroups of a group G is always a subgroup of G.

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.

Show answer
Correct Option: B
Explanation:

The union of two subgroups of a group G is a subgroup of G if and only if one subgroup is contained in the other.

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

Show answer
Correct Option: A
Explanation:

The set of all left cosets of H in G is denoted by G/H.

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

Show answer
Correct Option: D
Explanation:

The set of all right cosets of H in G is denoted by H\G.

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.

Show answer
Correct Option: C
Explanation:

The index of H in G is the number of cosets of H in G.

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.

Show answer
Correct Option: D
Explanation:

The normalizer of H in G is the set of all elements in G that satisfy any one of the given conditions.

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.

Show answer
Correct Option: A
Explanation:

The centralizer of H in G is the set of all elements in G that commute with every element in H.

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.

Show answer
Correct Option: B
Explanation:

A group G is called abelian if every element in G commutes with every other element in G.

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.

Show answer
Correct Option: A
Explanation:

A group G is called cyclic if it is generated by a single element.

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.

Show answer
Correct Option: A
Explanation:

The order of an element a in a group G is the smallest positive integer n such that $a^n = e$, where e is the identity element of G.

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.

Show answer
Correct Option: D
Explanation:

If the order of a subgroup H of a group G is a prime number p, then H is a normal subgroup of G, a cyclic subgroup of G, and every element in H has order p.

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.

Show answer
Correct Option: A
Explanation:

The Sylow theorems provide information about the existence and number of subgroups of a given order in a finite group.

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