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(ex) Consider the group G Ds (symmetries of a square) and the normal subgroup H-R,ro.iso.T2ro of this group. a. List the distinct left cosets of H in G b. Draw up an operation table for the group G/H e. By inspection, identify at least one familiar group that is isomorphic to G/H Definition: Suppose (G, ) is a group, with H a subgroup of G, and a G. Then a H denotes the set ashh e H and is called a left coset of H in G (or, if necessary, the left coset of H determined by a). (If the intended operation is clear, we usually denote aH by a, or even a+H if appropriate.) (To help you interpret this definition, note that it means that | E a * H iff there exists some h E H with z-as h.) 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) is called the quotient group of G by H. (Sometime the ter factor group" is used in place of "quotient group")

(ex) Consider the group G Ds (symmetries of a square) and the normal subgroup H-R,ro.iso.T2ro of this group. a. List the distinct left cosets of H in G b. Draw up an operation table for the group G/H e. By inspection, identify at least one familiar group that is isomorphic to G/H Definition: Suppose (G, ) is a group, with H a subgroup of G, and a G. Then a H denotes the set ashh e H and is called a left coset of H in G (or, if necessary, the left coset of H determined by a). (If the intended operation is clear, we usually denote aH by a, or even a+H if appropriate.) (To help you interpret this definition, note that it means that | E a * H iff there exists some h E H with z-as h.) 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) is called the quotient group of G by H. (Sometime the ter 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
Problem 1RQ
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(ex) Consider the group G Ds (symmetries of a square) and the normal subgroup H-R,ro.iso.T2ro of this group. a. List the distinct left cosets of H in G b. Draw up an operation table for the group G/H e. By inspection, identify at least one familiar group that is isomorphic to G/H Definition: Suppose (G, ) is a group, with H a subgroup of G, and a G. Then a H denotes the set ashh e H and is called a left coset of H in G (or, if necessary, the left coset of H determined by a). (If the intended operation is clear, we usually denote aH by a, or even a+H if appropriate.) (To help you interpret this definition, note that it means that | E a * H iff there exists some h E H with z-as h.) 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) is called the quotient group of G by H. (Sometime the ter factor group" is used in place of "quotient group")
Transcribed Image Text:(ex) Consider the group G Ds (symmetries of a square) and the normal subgroup H-R,ro.iso.T2ro of this group. a. List the distinct left cosets of H in G b. Draw up an operation table for the group G/H e. By inspection, identify at least one familiar group that is isomorphic to G/H Definition: Suppose (G, ) is a group, with H a subgroup of G, and a G. Then a H denotes the set ashh e H and is called a left coset of H in G (or, if necessary, the left coset of H determined by a). (If the intended operation is clear, we usually denote aH by a, or even a+H if appropriate.) (To help you interpret this definition, note that it means that | E a * H iff there exists some h E H with z-as h.) 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) is called the quotient group of G by H. (Sometime the ter factor group" is used in place of "quotient group")
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