# Is $H/N$ a quotient group even if $N$ is just a subgroup of $H$ and not necessarily normal?

I am very confused about the notation of $$H/N$$ and whether it always implies a quotient group.

The confusion stems from the following statement of the fourth isomorphism theorem:

Let $$G$$ be a group and $$N \unlhd G$$. Then every subgroup of the quotient group $$G/N$$ is of the form $$H/N =$$ {$$hn | h \in H$$} where $$N \leq H \leq G$$. Conversely, if $$N≤H≤G$$ then $$H/N≤G/N$$.

My question is: What does it mean by $$H/N$$ cause it just says $$N \leq H$$. Is $$H/N$$ a quotient group? I read that $$H/N$$ is a quotient group if and only if $$N \unlhd H$$, but nowhere in this theorem does it say that $$N$$ has to be a normal subgroup of $$H$$.

• Since $N$ is a normal subgroup of $G$, it is normal in any subgroup that contains it. – Clayton Apr 24 at 0:23
• You may check by definition: for any $h\in H$ one has $hN = Nh$, which is given by $N\lhd G$. – Hongyi Huang Apr 24 at 1:20
• Makes total sense. Thank you @Clayton. – Ufomammut Apr 24 at 2:09
• The notation $H/N$is **sometimes** used to denote the set of left cosets of $N$ in $H$, and not necessarily a group. However, the set of cosets is a group with the induced operation if and only if $N$ is normal in $H$ (though it need not be normal in $G$). Related: math.stackexchange.com/questions/14282/… – Arturo Magidin Apr 24 at 2:25
• Note, however, that if $N\triangleleft G$, and $N\leq H\leq G$, then $N\triangleleft H$. – Arturo Magidin Apr 24 at 2:26