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In this article the example of $\tfrac{50}{5} = 10$ example motivating quotient groups. I can see the idea, but I can't see any of the details really so I went & asked some friends, & some people in a math helproom & none of them see the connection...

Can anyone express the example of $50/5 = 10$ in quotient group terminology & notation, giving intuition on it & explaining some elementary results in quotient group theory using this example, as a way to allow me to always think back to this example anytime I want to quotient anything, whether it be a group, ring, topological space etc... Thanks very much

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So let $G$ be a group of order $50$ and $N$ a normal subgroup of order $10$. Then $G/N$ is a group of order $5$. That seems to be the only part missing in the connection to what he writes there. – Tobias Kildetoft Oct 24 '13 at 13:14
You say $50/5=10$ in the title but $50/10=5$ in the question... – lhf Oct 24 '13 at 14:23
Sorry, fixed it, thanks. – bolbteppa Oct 24 '13 at 16:12
up vote 1 down vote accepted

The connection is about the order of the group. As $50/5$ physically means creating $5$ sets with order $10$ out of the set with order $50$, algebraic meaning of Quotient Groups is just the same. If you find a normal subgroup $N$ in the group $G$, then order of $N$ should divide order of $G$, and what $G/N$ does is creating a partition of the elements of $G$.

Now let $G$ be a group in order $50$ and let $N$ be a normal subgroup of $G$ with order $10$. $G/N$ is a group containing $5$ elements where each contains $10$ elements. That's the same with the physical meaning of division, and it's actually the definition of division.

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Take a cyclic group $G$ of order $50$. Let $G=\langle g \rangle$. Let $H=\langle g^{5} \rangle$. Then $|H|=10$. The five cosets of $H$ in $G$ are given by $\{ g^{5q+r} : 0 \le q \le 9 \}$ for $0\le r \le 4$ and so $|G/H|=50/10=5$.

A concrete realization is $G=\mathbb Z/50 \mathbb Z$. Then $H=\mathbb 10Z/50 \mathbb Z \cong \mathbb Z/5 \mathbb Z$.

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