Understanding the free group of rank two is (as far as I know) very important to many different problems. I thought it would be helpful (for me and for others) to know more about it. Can anyone give me a list of interesting/useful properties of elements of $F_2$?
2026-03-26 11:01:36.1774522896
A list of properties of the free group of rank two
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Obviously, it has as subgroups the free groups of rank $1$ and $0$.
Nonobviously, it has as subgroups the free groups of all countable ranks. This follows from the fact that it's "SQ universal". That is, every countable group can be embedded in one of its quotients.
For a list of other interesting facts, there is always the Wikipedia article on free groups.