# Visualize left, right cosets and conjugation

I drew everything that's in orange.

Figure 6.8. Left illustration - Each left coset gH is where H arrows can reach from g, which looks like a copy of H based at g, as in the left illustration. Right illustration - Each right coset Hg is the set of nodes to which the g arrows take the elements of H.

Page 104 says --- The reason that left cosets look like copies of the subgroup while the elements of right cosets are usually more scattered is that I adopted the convention that arrows represent right multiplication. If I had used the convention where arrows represent left multiplication, right cosets would have been copies of the subgroup and left cosets would have been scattered.

Page 143 says --- Visually, $$gHg^{-1} = H$$ says all the $$g^{-1}$$ arrows lead back from the left coset gH to the subgroup $$H$$.

This is from Nathan Carter page 104 and Visual Group Theory.

Question 1. Where's the subgroup H in the left picture?
Question 2. Is my drawing right for gH? What's the circle around $$g$$?
Question 3. How do you see what page 143 says about 'all the $$g^{-1}$$ arrows lead back from the left coset gH to the subgroup $$H$$'?

EDIT @user901823 2/2/2014 ----

Question 4. What does "these pieces are permuted" mean in user901823's answer?
Question 5. 'Then when we form the factor group, this simply means we form a new group consisting of cosets (i.e. partition pieces) with group multiplication given by the permutations.'
I don't understand how permutations apply here. What permutations?
Question 6. What's $$g_1$$?
Question 7. How does $$g_1$$ multiplication 'shift H'? What does 'shift H' mean?

Question 1. Where's the subgroup H in the left picture?

It's not there. If it was, it would be a circle containing the identity $e$.

Question 2. Is my drawing right for gH? What's the circle around g?

Yes, it looks correct. The circle around $g$ is the coset $gH$.

Question 3. How do you see what page 143 says about 'all the $g^{−1}$ arrows lead back from the left coset gH to the subgroup H'?

The best way to think about cosets of a subgroup $H$ are as partitions of the group $G$ into equally sized pieces of size $|H|$. These pieces are permuted when multiplied on the right (or left) by group elements, with the condition that multiplying by an element of $H$ is the trivial permutation. So the $g_1$ arrows simply represent how $H$ has been shifted by $g_1$ multiplication.

Then when we form the factor group, this simply means we form a new group consisting of cosets (i.e. partition pieces) with group multiplication given by the permutations.

• thanks. I upvoted for you. can you please reply to my edit in my question? Can you please write in your answer and not in the comment box? – Group Theory Feb 2 '14 at 11:59