Our travel agency has put together an article about whether or not you can go to Hajj without going with a group via booking the Cheapest Hajj Packages. Of course, you can go to Haj without a group, and your Haj will still be valid. Now, if the person is a woman, she needs to be with a Mahram (which would be a male family member like husband, brother, father, Uncle, Grandfather). This is because the Haj takes about 5 days to finish, plus another week for travelling and waiting around. And a woman shouldn't stay alone in a place where millions of people have gathered.
What's the problem with that?
When there are a lot of strangers around, it's hard for a woman to eat, relax, sleep, and do other things. The other part of your question is that there are limits on how many people from each country can go on Haj. And they travel in groups because there are challenges like food, language, asking for directions, and first aid. A woman can't go somewhere without a mahram with her. A mahram is a man she can't marry for the rest of her life, either because they are related by blood or for another reason. This also depends on him being mentally healthy as an adult. A mahram can't be a small child or someone who isn't in their right mind.
A mahram is needed so that he can watch over and protect her, and so that she doesn't have to deal with people who don't fear Allah SWT and don't treat His slaves with kindness. It doesn't matter if she is with other women or alone, and it doesn't matter if she is in danger or not. She can't go anywhere without a mahram, even if she goes with other women from her own family and feels safe. Because when the Prophet (peace and blessings of Allah be upon him) told the man to go to Hajj with his wife, he didn't ask if there were any other women with her or if she was safe.
He didn't ask about it, which shows that he didn't think it was important. This is the right way to look at it. Some people today don't think much about the problem and let women fly alone without a mahram. This is definitely not in line with what the passages seem to say. Traveling by plane is risky, just like travelling in any other way. If a woman's mahram drives her to the airport, he must leave her there as soon as she gets to the departure hall. If he doesn't, the woman will be on her own without a mahram. The plane could arrive on time or late. It might leave on time but then have to come back, or it might land at a different airport than the one she wanted to go to, etc.
It's possible that it won't get to the airport where she's flying on time. If it gets there on time, the mahram who is supposed to meet her could be late for a number of reasons, such as oversleeping, getting stuck in traffic, having his car break down, or something else. The idea is that women should be afraid of Allah SWT and only go to Hajj or anywhere else with a mahram who is an adult and has a clear head. And what we want is for Allah SWT to help us.