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(ex) Consider the group (Z, +), and its cyclic subgroup H - (5) (n e Zn 5m for some m edenoted earlier by 52). List the distinct left cosets of H in (Z, +), and list the elements of each of these. Howmany distinct left cosets are there for H?Photos -cosets.PNGCosets: First Ideas and Applicationsin mathematics there is a way of looking at things that at first seems rather useless, but which turns outto bereally powerful. Cosets are like that. At first it might seem that this is a strangely irrelevant idea to spendtime 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 zThe following exercises should help you to understand this definitiona * h.

Question

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.
help_outline

Image Transcriptionclose

(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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Step 1

Given H = <5> = {n∈ Z | n = 5m, for some integer m}.

The left cosets will be:

0 + H, 1 + H, 2 + H, 3 + H, 4 + H,... and in general a + H, where a is any integer.

Step 2

To find the distinct left cosets.

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Step 3

To list the 5 distinc...

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