# (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.

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.

Step 3

To list the 5 distinc...

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