Should I let my daughter shave her head?

Okay, so my daughter Alex wants to shave her head. Her best friend Brittany was diagnosed with Lukemia a few weeks ago. She's started chemotherapy and her hair has begun to fall out(which I know must be terrible for Brittany. Her hair is yellow-blong and straight and she's grown it about three feet long). Anyway, Alex wants to shave her head to show her support to Brittany. They are very close. Her grandmother (who she lives with as her parents are dead) is a good friend of mine and the girls agree on nearly everything. They've been friends sice the diaper days. I know why she wants to do thin, but still. She's eleven, she might regret it. And her hair is so beautiful. Not as pretty as Brittany's maybe, but it's red and goes past her shoulders and is so curly. Alex wants to have it made into a wig for Brittany and show her support to her, but I still aren't sure I should let her in case it looks terrible in the end when she's bald. And while Brittany would be allowed to wear a hat in class next year because her baldness is because of a disease, that wouldn't apply for Alex if it took a long time for her hair to grow. Should I let her, or will she regret it if she does it?