Intuitive meaning of left and right cosets?

What is the difference between left and right cosets? I know their definition, but what I am seeking is the intuition behind left and right coset.

I used to think of cosets as slicing a group (since I am learning group theory) into different segments that has no intersection. However, why then is there left and right cosets?

I came across this thing that I found very puzzling if $gH = Hk$ for some $k,g$ in group $G$ then $gH = Hg$. I prove it using $g1_H = hk$ so $1_Hg =hk$ and hence Hg = Hk = gH. I cannot get the intuition behind it. Can someone enlighten me please.

The set $gH$ is the set $\{gh \colon h \in H\}$ while $Hg$ is the set $\{hg \colon h \in H\}$. In general these two sets are just different things.

You can chop-up a group into right co-sets and you can chop-up a group into left co-sets. In general this yields different partitions.

Sometimes they yield the same thing, and this is then remarkable.

What I want to get across is that a priori there is no reason why the two things should (ever) be the same. So, one considers both. Sometimes, they coincide and then one can forget about the distinction.

Consider $G=S_3$ the permutation group on three elements. And $H$ the subgroup generated by $(1,2)$.

Then

• $(1,3)H= \{(1,3) , (1,2,3)\}$, while $H(1,3)= \{(1,3) , (1,3,2)\}$.
• $(2,3)H= \{(2,3) , (1,3,2)\}$, while $H(2,3)= \{(2,3) , (1,2,3)\}$.

Only

• $(1,2)H= H= H(1,2)$.

Thus you get two distinct partitions of $S_3$. For the one you get

$$\{ (1), (1,2)\} \cup \{(1,3) , (1,2,3)\} \cup \{(2,3) , (1,3,2)\}$$ for the other you get $$\{ (1), (1,2)\} \cup \{(1,3) , (1,3,2)\} \cup \{(2,3) , (1,2,3)\}$$

Thus you have two distinct partitions depending on your choice.

And, for your specific last question, what this says is that if a left coset associated to some $g$ is a right coset at all then it is already the right coset associated to the same element $g$.

You can see this in the example above $(1,2)H = H(1,2)$ where as the other right cosets are not left cosets at all, and vice versa.