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We can intuitively understand a normal subgroup by visualizing the group G as an outer rectangle and the subgroup H as an inner rectangle. The condition $gHg^{-1} = H$ means that if you 'transport' the entire region H using an element g and then bring it back using $g^{-1}$, the resulting region still lies completely within H.

Conditions: H is a subgroup of G; g is an element of G outside H

We can intuitively understand a normal subgroup by visualizing the group G as an outer rectangle and the subgroup H as an inner rectangle. The condition $gHg^{-1} = H$ means that if you 'transport' the entire region H using an element g and then bring it back using $g^{-1}$, the resulting region still lies completely within H.

Conditions: H is a subgroup of G; g is an element of G outside H