question asked around a weird concept 
I am struggling with these questions. I dont know what is meant by the term system of representatives.
anybody knows about these things?
 A: The quotient group $G/H$, where $G$ is a group and $H$ is a normal subgroup, is the set of equivalence classes under the equivalence relation
$$
a\sim_H b \qquad\text{if and only if}\qquad ab^{-1}\in H
$$
It is well known that the equivalence classes are the subsets of the form
$$
aH=\{ah:h\in H\}
$$
as $a\in G$, called cosets. More precisely left cosets, because one can also consider $Ha$ (right coset). However, in case $H$ is a normal subgroup, $aH=Ha$.
A system of representatives is a subset $X\subseteq G$ such that

for each $a\in G$, there exists one an only one $x\in X$ such that $a\sim_H x$.

If $G=\langle a\rangle$ is cyclic of order $n$ and $H=\langle a^d\rangle$, where $d$ divides $n$, then $a^d$ has order $n/d$, so the quotient group, by Lagrange's theorem, has order $d$. Note that $G/H$ is cyclic, generated by $aH$ (why?) and the set we might think to is
$$
X=\{1=a^0,a=a^1,a^2,\dots,a^{d-1}\}.
$$
Is this set a system of representatives? Hint: it is sufficient to prove that $a^i\not\sim_H a^j$ (that is, $a^i(a^j)^{-1}\notin H$) for $0\le i<d$, $0\le j<d$ and $i\ne j$.
Similar considerations apply to $\mathbb{R}/\mathbb{Z}$, with the difference that $a+\mathbb{Z}$ is used for denoting a coset.
Hint: for $r\in\mathbb{R}$, consider its floor $\lfloor r\rfloor$ (the largest integer $n$ such that $n\le r$) and $r-\lfloor r\rfloor$, sometimes named the *fractional part of $r$.
A: There's a canonical map $\pi: G \to G/H$ which is usually defined by $\pi(g) = g+H$ and is sometimes written as $\bar g$. A system of representatives is then a set $S \subset G$ such that for every element $x\in G/H$ there is precisely one element $g\in G$ with $\pi(g) = x$. For example, take $G = \mathbb Z$ and $H = 6\mathbb Z$ then I could set $S = \{0, 1, 2, 3, 4, 5\}$ or something crazier like $S = \{6, -5, 26, 45, 4, -1\}$ and both are systems of representatives, since both are mapped bijectively into $\mathbb Z / 6\mathbb Z$ by $\pi$. Equivalently, $S$ contains precisely one element from every coset, so in the example you can verify that there is exactly on element from each of the cosets $6\mathbb Z, 6\mathbb Z + 1 ,\dots, 6\mathbb Z + 5$.
