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(ex) Consider the group (Z, +), and its cyclic subgroup H - (5) (n e Zn 5m for some m e denoted earlier by 52). List the distinct left cosets of H in (Z, +), and list the elements of each of these. How many distinct left cosets are there for H? Photos -cosets.PNG Cosets: First Ideas and Applications in mathematics there is a way of looking at things that at first seems rather useless, but which turns out to be really powerful. Cosets are like that. At first it might seem that this is a strangely irrelevant idea to spend time on. Bear with this: you will soon find how much can be derived from the idea. Definition: Suppose (G, ,) is a group, with H a subgroup of G, and a e G. Then a * I denotes the set fa hlh 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 a H by aH, or even a+ H if appropriate.) To help you interpret this definition, note that it means that z e a * H iff there exists some h e H with z The following exercises should help you to understand this definition a * h.

(ex) Consider the group (Z, +), and its cyclic subgroup H - (5) (n e Zn 5m for some m e denoted earlier by 52). List the distinct left cosets of H in (Z, +), and list the elements of each of these. How many distinct left cosets are there for H? Photos -cosets.PNG Cosets: First Ideas and Applications in mathematics there is a way of looking at things that at first seems rather useless, but which turns out to be really powerful. Cosets are like that. At first it might seem that this is a strangely irrelevant idea to spend time on. Bear with this: you will soon find how much can be derived from the idea. Definition: Suppose (G, ,) is a group, with H a subgroup of G, and a e G. Then a * I denotes the set fa hlh 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 a H by aH, or even a+ H if appropriate.) To help you interpret this definition, note that it means that z e a * H iff there exists some h e H with z The following exercises should help you to understand this definition a * h.

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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Abstract Algebra

I need help on a problem that requires the definition of Cosets.

The definition is given in the given picture as well as the problem above it.

 

(ex) Consider the group (Z, +), and its cyclic subgroup H - (5) (n e Zn 5m for some m e denoted earlier by 52). List the distinct left cosets of H in (Z, +), and list the elements of each of these. How many distinct left cosets are there for H? Photos -cosets.PNG Cosets: First Ideas and Applications in mathematics there is a way of looking at things that at first seems rather useless, but which turns out to be really powerful. Cosets are like that. At first it might seem that this is a strangely irrelevant idea to spend time on. Bear with this: you will soon find how much can be derived from the idea. Definition: Suppose (G, ,) is a group, with H a subgroup of G, and a e G. Then a * I denotes the set fa hlh 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 a H by aH, or even a+ H if appropriate.) To help you interpret this definition, note that it means that z e a * H iff there exists some h e H with z The following exercises should help you to understand this definition a * h.
Transcribed Image Text:(ex) Consider the group (Z, +), and its cyclic subgroup H - (5) (n e Zn 5m for some m e denoted earlier by 52). List the distinct left cosets of H in (Z, +), and list the elements of each of these. How many distinct left cosets are there for H? Photos -cosets.PNG Cosets: First Ideas and Applications in mathematics there is a way of looking at things that at first seems rather useless, but which turns out to be really powerful. Cosets are like that. At first it might seem that this is a strangely irrelevant idea to spend time on. Bear with this: you will soon find how much can be derived from the idea. Definition: Suppose (G, ,) is a group, with H a subgroup of G, and a e G. Then a * I denotes the set fa hlh 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 a H by aH, or even a+ H if appropriate.) To help you interpret this definition, note that it means that z e a * H iff there exists some h e H with z The following exercises should help you to understand this definition a * h.
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