Today in math my professor was talking about equivalence relations and modulo. Then he said something about cosets, and I don't understand the relation. I read that a coset was a subgroup with all if it's elements added with another element of the group. And I guess equivalence relations for subgroups, at least the integers mod some number? My main question is how are cosets and equivalence relations related?
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See https://math.stackexchange.com/a/1014784/589 – lhf Apr 01 '23 at 10:29
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The right cosets of a subgroup $H\le G$ are precisely the classes of the equivalence relation "right modulo $H$" in $G$ defined by: $$x\sim_R y\iff xy^{-1}\in H$$ In fact, $xy^{-1}\in H\iff x\in Hy$, and hence: $$[y]_R=Hy$$ Note that the left cosets are instead the classes of the equivalence relation "left modulo $H$": $$x\sim_L y\iff y^{-1}x\in H$$ namely: $$[y]_L=yH$$ In an abelian group the two equivalence relations are actually one and the same, as $xy^{-1}=y^{-1}x$ for every $x,y\in G$. In particular, for $G=(\Bbb Z,+)$ and $H=n\Bbb Z$ (for any $n\in\Bbb N$), the cosets are the sets: \begin{alignat}{1} [m]_n&=m+n\Bbb Z \\ &=\{m+kn\mid k\in \Bbb Z\} \\ \end{alignat} for any $m\in \Bbb Z$. Note that: $$[m]_n=[m+n]_n$$
citadel
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Cosets and equivalences relations are not related, except inasmuch as you can define an equivalence relation fairly naturally on the cosets of some subgroup of a group.
For example, taking the group $G = (\mathbb{Z}, +)$, and its subgroup $(3\mathbb{Z},+)$ gives you an infinite number of cosets $(\dots,-1 + 3\mathbb{Z}, 0 + 3\mathbb{Z}, 1 + 3\mathbb{Z}, 2 + 3\mathbb{Z}, 4 + 3\mathbb{Z},\dots)$. However, as $1 + 3\mathbb{Z}$ and $4 + 3\mathbb{Z}$ define the same sets, they are equivalent, both being the set of numbers equivalent to $1$ mod $3$.
qaphla
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