Normal subgroup

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A subgroup N of a group G is a normal subgroup (written $\text{[math]}$ ) if one of the following equivalent conditions holds:

$\text{[math]}$ for all $\text{[math]}$
$\text{[math]}$ for all $\text{[math]}$
$\text{[math]}$ for all $\text{[math]}$

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Algebra