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Prove that if N is a subgroup of G, and the operation (Na)(Nb) = N(ab) is well defined

Chapter 22, Problem 22.15

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QUESTION:

Prove that if N is a subgroup of G, and the operation (Na)(Nb) = N(ab) is well defined onthe set G / N of all right cosets of N in G, then N

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QUESTION:

Prove that if N is a subgroup of G, and the operation (Na)(Nb) = N(ab) is well defined onthe set G / N of all right cosets of N in G, then N

ANSWER:

Step 1 of 2

Let us considerbe a group,be a subgroup of the group.

The well-defined operation is:

for all 

Suppose that is the set of all right cosets ofis.

To show thatis a normal subgroup of, claim thatforand.

Consider a closet if:

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