Questions about cosets, conjugate classes etc Some questions about subgroups, normal subgroups, conjugate classes etc, just to make sure I understand it :-)


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*The index of a subgroup $H$ in $G$, written as $[G:H]$ is defined as the number of left cosets of $H$ in $G$. I know that the a left coset of $H$ in $G$ is determined by $a$, when $$ aH= \{x\in G ; x=ah, h\in H\}$$ 
So my questions about this thing are: 
1.) Are the left cosets of $H$ in $G$ disjunct? Why?
2.) Can we say that $$G = \bigcup_{x\in G} xH$$so $G$ is the union of all different left cosets in $G$? How can I see/prove this?
3.) I read $ \#G= [G:H]\cdot\#H$. Why is this? Is there any relation between this fact and Lagrange theorem?
4.) What about the right cosets? I guess the union of all the distinct right cosets is the same set as the union of left cosets? What can one say about the relation between left and right cosets?
 A: Questions 1 and 2
Define a relation on $G$ by $x\sim_H y$ if and only if $x^{-1}y\in H$. This is an equivalence relation. Can you prove it?
What are the equivalence classes? The class of the element $x$ is the set of all elements $y\in G$ such that $x^{-1}y\in H$ which is easily seen to be $xH$. Thus the equivalence classes are exactly the (left) cosets. This implies, by general theory of equivalence relations, that the cosets are non empty, pairwise disjoint and their union is $G$.
Question 3
Define the map $\varphi\colon H\to xH$ by $\varphi(h)=xh$. This is a bijection, therefore all cosets share the same cardinality of $H$ (which is the coset $1H$, by the way). In case $G$ is finite, this has the consequence that
$$
|G|=|H|\cdot [G:H].
$$
Just count the elements, recalling that the union of the cosets is $G$.
Question 4
Define a relation on $G$ by $x\mathrel{{}_H\!\sim} y$ if and only if $yx^{-1}\in H$. Repeat the same reasoning as above, the only difference is that the equivalence classes are the right cosets.
Question 5
If $[G:H]=2$, then we know we have two distinct left cosets: $H$ and $xH$, where $x\in G\setminus H$. But, since $x\in G\setminus H$, the right coset $Hx$ is different from $H$. The cosets (left or right) define a partition of $G$: hence $xH=G\setminus H=Hx$, so any left coset is also a right coset.
Question 6
Yes.
Question 7
The usual notation is $G/N$. The case $\mathbb{Z}/n\mathbb{Z}$ is just a particular case (the only difference is that the operation is denoted by $+$).
