[Math] Intuition of coset of a subgroup

abstract-algebragroup-theory

Hey guys I am trying to form the intuition that distinct left coset of subgroups are actually disjoint. I understand the proof constructed but I don't think I get the intuition behind why that the case if someone could maybe explain to me that would be great. I think this is very important to understand concretely since Lagrange theorem which has very intuitive proof is based on it.

Definition of coset:

Let H be a subgroup of the group G. For any $a \in G$,

$aH =\{x \in G : x = ah \quad \text{for some} \quad h \in H\}$
is a left coset of $H \in G$.
Similarly, $Ha$ is called a right coset of $H \in G$.

Best Answer

A nice way to think about cosets is through equivalence classes.

Intuitively, if you have an equivalence relation $\sim$ on a set $X$, then this equivalence relation "lumps" the set $X$ into smaller sets, called equivalence classes. A subset $A\subset X$ is an equivalence class if there is some $a \in A$ such that $A$ is the set of all $x \in X$ satisfying $x \sim a$. A really simple example is when the set is $\mathbb{R}$ and $\sim$ is the usual equality of numbers, $=$. In this case, any equivalence class $A$ is just a singleton; $x \sim y$ for $x,y \in A$ means $x=y$. In this simple instance, $=$ "partitions" $\mathbb{R}$ into a bunch of classes (the singletons) for which each $x \in \mathbb{R}$ belongs to exactly one.

In the case of cosets, we can consider a subgroup $H\subset G$ (with identity $e$) and define a relation $\sim$ on $G$ by saying that $x\sim y$ if and only if $xy^{-1} \in H$. Indeed, this is an equivalence relation. Note the following:

  • We have that $xx^{-1} = e \in H$, so $x\sim x$.

  • If $x\sim y$, then $xy^{-1} \in H$, so $yx^{-1} = (xy^{-1})^{-1} \in H$ since $H$ is closed under inverses;

  • If $x\sim y$ and $y \sim z$, then $xy^{-1} \in H$ and $yz^{-1} \in H$. As $H$ is closed under the group operation, $(xy^{-1})(yz^{-1}) = xz^{-1} \in H$. So, $x\sim z$.

These three properties are the necessary conditions for $\sim$ to be an equivalence relation. Now, we can ask, what are the corresponding equivalence classes? Well, we will see that they are sets $A\subset G$ for which $x,y\in A$ means that $xy^{-1} \in H$. That is, if we fix an $a\in A$, then elements $x\in A$ have the property that $xa^{-1} = h$ for some $h \in H$. But then $x = ha$, and since $x \in A$ was arbitrary, we can then identify $A$ with $Ha$, which is your definition of a right coset. Left cosets arise in a very similar manner by considering the equivalence relation $x\sim y$ when $x^{-1} y \in H$ (note the difference in order!).

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