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$G=(Z_{20},+_{20})$ and $H=\langle [4]\rangle$. List the distinct right coset of $G$. What is order of $H+[6]$? Is $\frac{G}{H}$ isomorphic to $Z_4$? How to find right coset? Thanks in advance.

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Have you googled for "Klein four group"? –  Mariano Suárez-Alvarez Apr 24 '12 at 6:04
It's the group of order 4 in which every element has order 2.... –  user38268 Apr 24 '12 at 6:04
Or the vector space with four elements ;) –  Bruno Joyal Apr 24 '12 at 6:07

1 Answer 1

up vote 2 down vote accepted

If $G$ is a group and $H$ is a subgroup of $G$ and $g$ is an element of $G$ then the right coset of $H$ in $G$ containing $g$ is the set of all elements of the form $gh$ where $h$ is in $H$. [or else it's the set of all elements $hg$ - it depends on what text you are using]

So, you find a right coset by picking an element $g$ and calculating $gh$ for every $h$ in $H$.

If the group operation is written as addition instead of multiplication, then you calculate $g+h$ for every $h$ in $H$.

If you want all the right cosets, you have to do this calculation for various different elements $g$ until you get enough cosets to include all of $G$.

Try it!

Now that you have found all the right cosets (and $G/H$ is simply notation for the set of all the cosets), do you know how to add two of them to get a right coset? It's easy: (right coset containing $a$) plus (right coset containing $b$) equals (right coset containing $a+b$). (note that now I'm assuming the operation is written as addition)

So now that you have all the cosets, and you know how to add two to get a coset, you can write out the addition table for the cosets, the table that systematically gives the result of all possible additions of two cosets. Then you can look at that table, and look at the table for ${\bf Z}_4$ (I take it you know what that table looks like, or at least how to construct it), and see whether the two tables have the same structure. Then you will know whether the two groups are isomorphic.

And in case this is not clear, I'm sure your teacher will be happy to help you.

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