C. Elements That Commute
If a and b are in G and ab = ba, we say that a and b commute. Assuming that a and b commute, prove the following:
1. a^?1 and b^?1 commute.
2. a and b^?1 commute. (Hint: First show that a = b^?1ab.)
3. a commutes with ab.
4. a^2 commutes with b^2.
5. xax^?1 commutes with xbx^?1, for any x in G.
6. ab = ba iff aba^=1 = b.
(The abbreviation iff stands for “if and only if.” Thus, first prove that if ab = ba, then aba^?1 = b. Next, prove that if aba^?1 = b, then ab = ba. Proceed roughly as in Exercise A. Thus, assuming ab = ba, solve for b. Next, assuming aba^?1 = b, solve for ab.)
7. ab = ba iff aba^?1b^?1 = e.
