Find the quotient group(space) of $\Bbb Z/H$ if $H = 6\Bbb Z $ and it is also a subgroup of $\Bbb Z$.
Do the same if $ H = \langle[4]\rangle $ in $ \Bbb Z_{12}$
The quotient group of $\Bbb Z/H$
$H = \{...,-12, -6, 0, 6, 12,...\}$
I am not quite sure what is the quotient group of $\Bbb Z/H.$
I think it is $\{[6]\}$. I only know about quotient group that is related to equivalence relation.
For $H = \langle[4]\rangle $ in $ \Bbb Z_{12}$,
$[4] = \{4, 16, 20, 24,...\}$
$\langle[4]\rangle = \{4\}$ since the remainders of $4^k$, with $k \in Z$ are always 4.
Again, I am not sure what the quotient space is here.
I am in abstract algebra and we haven't talked about topology yet.
"The quotient group of $\Bbb Z/H$
$H = \{...,-12, -6, 0, 6, 12,...\}$
I am not quite sure what is the quotient group of $\Bbb Z/H.$ "
It is the cosets of $H$ in $\mathbb{Z}$
$\Bbb Z/H = \{0+H, 1+H, 2+H, 3+H, 4+H, 5+H \} \cong \{0, 1, 2, 3, 4, 5\}$
with addition modulo $6$.