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<p>generated by the set-theoretic union of the subgroups, not the set-theoretic union itself.) If  is the identity of , then the trivial group  is the <a href="page.php?w=partial_order">minimum</a> subgroup of , while the <a href="page.php?w=partial_order">maximum</a> subgroup is the group  itself.</p>

<p><big>Cosets and Lagrange's theorem</big></p>
<p>Given a subgroup  and some  in , we define the <b>left <a href="page.php?w=coset">coset</a></b>  Because  is invertible, the map   given by  is a <a href="page.php?w=bijection">bijection</a>. Furthermore,</p><p>
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