# Alg7.17

From Example Problems

Let be a group such that for all and for three consecutive integers . Prove is abelian.

We are given three equations. For all

(1)

(2)

(3)

From (1), we have

.

From (2), by multiplying by the inverse found from (1), we have

(4)

From (3), again multiplying by the inverse found from (1), we have

(5)

Take (4) and mutliply by on the left and (5) and multiply by on the left and we get

Now cancelling the on the right, we get

Therefore, for every , it is true that commutes with every element of . Using that information and (4), we get

Therefore, for every element of commutes with every other element of .