**Motivation For Definition**

The reason *G*/*N* is called a quotient group comes from division of integers. When dividing 12 by 3 one obtains the answer 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, however we end up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects.

To elaborate, when looking at *G*/*N* with *N* a normal subgroup of *G*, the group structure is used to form a natural "regrouping". These are the cosets of *N* in *G*. Because we started with a group and normal subgroup the final quotient contains more information than just the number of cosets (which is what regular division yields), but instead has a group structure itself.

Read more about this topic: Quotient Group

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