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Are the following statements true or false?

  1. Let $M$ be a set and $\sim$ be an equivalence relation in $M$. Then every equivalence class has the same number of elements.
  2. Let $(G, \circ)$ be a group and $H \subset G$ a subgroup. Then every equivalence class in the factor group $G/H$ has the same number of elements.

For 1. I suppose it is false, since if we let $M=\{\{1\},\{2\},\{2,3\}\}$ and $\sim$ is "having the same number of elements". Then $|[\{1\}]|=2$, but $|[\{2,3\}]|=1$. Is this right?

For 2. I have a strong feeling that it is true, however I am unable to prove it. Any hints welcome.

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    Hint for 2.: for $g \in G$, can you build a bijection between $H$ and the class $gH = \{gh : h \in H\}$? – 2017-01-19
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    Of course $M$ is a finite set and $(G,\circ)$ is a finite group. Note that if you are successful in proving 2°), you will get the famous Lagrange theorem which asserts that the cardinal (= number of elements) of any subgroup of a finite group divides the cardinal of the group itself. – 2017-01-19
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    $\newcommand{\Set}[1]{\left\{ #1 \right\}}$For (1) you are right, but you may make things simpler by building directly the partition, and recovering from it the equivalence relation. So for instance the partition $\mathfrak{P} = \Set{ \Set{1}, \Set{2, 3}}$ of $A = \Set{1, 2, 3}$ yields the equivalence relation on $A$ given by $a \sim b$ iff there is $P \in \mathfrak{P} $ such that $a, b \in P$, whose equivalence classes are precisely the elements of $\mathfrak{P}$. – 2017-01-19

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