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4. a. Suppose G is a group, and H is a normal subgroup of G. Prove or disprove as appropriate: If G is abelian then G/H is abelian b. Suppose G is a group, and H is a normal subgroup of G. Prove or disprove as appropriate: I G/H is abelian, then G is abelian Definition: A subgroup H of a group G is said to be a normal subgroup of G if for all ae G, aH-Ha Definitions: . A group (G,) is said to be abelian if is commutative. We say a group is finite if the underlying set contains finitely many elements. We say a group is infinite if the underlying set contains infinitely many elements For a finite group G, the order of G is the number of elements in G Definition: Suppose G is a group, and H a normal subgroup of G. The group consisting of the set G/H with operation defined by (aH) (bH) (ab)H is called the quotient group of G by H. (Sometime the term "factor group" is used in place of "quotient group)

4. a. Suppose G is a group, and H is a normal subgroup of G. Prove or disprove as appropriate: If G is abelian then G/H is abelian b. Suppose G is a group, and H is a normal subgroup of G. Prove or disprove as appropriate: I G/H is abelian, then G is abelian Definition: A subgroup H of a group G is said to be a normal subgroup of G if for all ae G, aH-Ha Definitions: . A group (G,) is said to be abelian if is commutative. We say a group is finite if the underlying set contains finitely many elements. We say a group is infinite if the underlying set contains infinitely many elements For a finite group G, the order of G is the number of elements in G Definition: Suppose G is a group, and H a normal subgroup of G. The group consisting of the set G/H with operation defined by (aH) (bH) (ab)H is called the quotient group of G by H. (Sometime the term "factor group" is used in place of "quotient group)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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4. a. Suppose G is a group, and H is a normal subgroup of G. Prove or disprove as appropriate: If G is abelian then G/H is abelian b. Suppose G is a group, and H is a normal subgroup of G. Prove or disprove as appropriate: I G/H is abelian, then G is abelian Definition: A subgroup H of a group G is said to be a normal subgroup of G if for all ae G, aH-Ha Definitions: . A group (G,) is said to be abelian if is commutative. We say a group is finite if the underlying set contains finitely many elements. We say a group is infinite if the underlying set contains infinitely many elements For a finite group G, the order of G is the number of elements in G Definition: Suppose G is a group, and H a normal subgroup of G. The group consisting of the set G/H with operation defined by (aH) (bH) (ab)H is called the quotient group of G by H. (Sometime the term "factor group" is used in place of "quotient group)
Transcribed Image Text:4. a. Suppose G is a group, and H is a normal subgroup of G. Prove or disprove as appropriate: If G is abelian then G/H is abelian b. Suppose G is a group, and H is a normal subgroup of G. Prove or disprove as appropriate: I G/H is abelian, then G is abelian Definition: A subgroup H of a group G is said to be a normal subgroup of G if for all ae G, aH-Ha Definitions: . A group (G,) is said to be abelian if is commutative. We say a group is finite if the underlying set contains finitely many elements. We say a group is infinite if the underlying set contains infinitely many elements For a finite group G, the order of G is the number of elements in G Definition: Suppose G is a group, and H a normal subgroup of G. The group consisting of the set G/H with operation defined by (aH) (bH) (ab)H is called the quotient group of G by H. (Sometime the term "factor group" is used in place of "quotient group)
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