Did your daughter enter a new school this year? Our 12-year-old son is an only and always seems to have a hard time with transitions. I think that is more just him than a matter of him being an only.
When he was younger, I used to arrange a lot of play dates and signed him up for various sports, usually one per season, being careful to not overschedule him. (It drives me nuts when people have their kids scheduled from morning till night.) But as he has gotten older, he has dropped out of some of those activities. He still keeps up with Scouts, weekend youth group that meets twice a month, and an occasional sport.
Part of me worries that our son is lonely at times, but we learn things about ourselves during those times as well. I have to be careful to let him feel the aloneness as well as the camaraderie of friends and family.
At 10, he had a harder time making friends on his own. He was past the "play dates" I arranged but didn't have the confidence to create his own. It took some nudging of encouraging him to make the phone calls, but he has grown more comfortable with it through time. Sometimes practicing the wording before the call is made helps. Sometimes play dates of a sort are still facilitated by me. But, in general, our son has become much more willing to take his own steps toward orchestrating his own get togethers.
It's painful to watch someone feel lonely sometimes. But it also can build the desire to do something about it.
Good luck!