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A normal subgroup $N$ is a subgroup where the left cosets are the same as the right cosets. $N$ is normal $\iff $ $xnx^{-1} \in N, \forall x\in G$.
5.) Why is it that if $[G:H]=2 \implies $ $H$ is normal subgroup?
6.) Can we say that a factor group is just a group that has left cosets of $N$ (being a normal subgroup) as its elements? So if $N$ is a normal subgroup, then the left cosets of $N$ forms a group under coset multiplication given by $aNbN = abN$.
7.) The group of left cosets of $N$ in $G$ is called the factor group, why do we denote this by $G/N$? These are the same things as the integers modulo $n$ groups? How can I relate those exactly?
Best Answer
$(5)$ If $\;[G:H] = 2,\,$ then by definition of the index of $H$ in $G$, there exists exactly two distinct left cosets: $H$ and $gH$, for $g \in G \setminus H.\;$ Now, for $g\in G\setminus H$, the right coset $Hg \neq H$. We know the left cosets of $H$ partition $G$, as do the right cosets of $G$. It follows, necessarily, that $gH=G\setminus H=Hg$, so any left coset is also a right coset in any subgroup of index $2$
For $(6)$: Yes, indeed. Spot on.
$(7)$ Yes, the usual notation for the factor group (also sometimes called a quotient group) is $G/N$. The integers modulo $n = \mathbb{Z}/n\mathbb{Z}\,$ which is simply a particular example of such a group, where the group $G = \mathbb Z$ and $N = \mathbb n\mathbb Z$.