This phase III trial compares the effectiveness of high dose chemotherapy and the patients' own (autologous) stem cells to observation only in patients with peripheral T-cell lymphoma who achieved a complete response after initial chemotherapy. Participants will be separated into two groups, or randomized (like flipping a coin). The first group of participants will receive high dose chemotherapy followed by an autologous stem cell transplant. Giving chemotherapy before a stem cell transplant helps kill cancer cells in the body and helps make room in the patient's bone marrow for new blood-forming cells (stem cells) to grow. Stem cells removed prior to treatment are then returned to the patient to replace the blood forming cells that were destroyed by the chemotherapy. The second group will be observed by the study doctor and study team with check-ins every six months for 12 years after joining the trial. The group of participants who receive a stem cell transplant will also be checked by their doctor and study team for every 6 months for 12 years after joining the study. The main risks of the study treatment include reduced blood cell counts, nausea and vomiting, and damage to other organs (due to high dose chemotherapy).