Normal subgroups and factor groups

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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?

Solution 1:

$(5)$ 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 $N = \mathbb n\mathbb Z$.

Solution 2:

For 5: We have two left cosets, $H$ and $xH\ne H$. We have two right cosets, $H$ and $Hy$. Since $Hy$ is disjoint from $H$, it must be $xH$, thus left and right cosets of $H$ are the same.

Solution 3:

The choice of numbering seems curious. For 7, looking at the integers (mod n) is the special case where $G = (\mathbb{Z},+)$ and $N$ is the additive subgroup $n\mathbb{Z}.$ The analogy is somewhat obscured by the fact that the group operation is addition with integers (mod $n$) and in the general case the operation is usually multiplication (or just juxtaposition). Also, in the special case, the whole group is Abelian which makes the construction much more straightforward than the general case.