From wikipedia:
In set theory, a hereditary set (or pure set) is a set all of whose elements are hereditary sets. That is, all elements of the set are themselves sets, as are all elements of the elements, and so on.
So, sets like $\emptyset,\ \{\emptyset\},\ \{\{\emptyset\}\},\ \{\{\emptyset\},\ \emptyset\}\ ...$
It's easy to see that if you take the set $S$ of all the finite pure sets (whose elements are also finite pure sets) and equip it with the operation of symmetric difference $A \triangle B = \left(A\cup B\right) \setminus \left(A\cap B\right) = \left(A\setminus B\right)\cup\left(B\setminus A\right)$ you obtain an infinite group.
My question is, how do I approach this group to understand its structure? So far I've only found out some of its subgroups (free abelian groups on $n$ elements where each element has order 2), but I'm sure there's much more to it