Suppose a group ${\displaystyle G}$  acts on a set ${\displaystyle A}$ . Then ${\displaystyle A}$  is ${\displaystyle G}$ -paradoxical if there exists some disjoint subsets ${\displaystyle A_{1},...,A_{n},B_{1},...,B_{m}\subseteq A}$  and some group elements ${\displaystyle g_{1},...,g_{n},h_{1},...,h_{m}\in G}$  such that:^[1]

${\displaystyle A=\bigcup _{i=1}^{n}g_{i}(A_{i})}$  and ${\displaystyle A=\bigcup _{i=1}^{m}h_{i}(B_{i})}$ 

Free group edit

The Free group F on two generators a,b has the decomposition ${\displaystyle F=\{e\}\cup X(a)\cup X(a^{-1})\cup X(b)\cup X(b^{-1})}$  where e is the identity word and ${\displaystyle X(i)}$  is the collection of all (reduced) words that start with the letter i. This is a paradoxical decomposition because ${\displaystyle X(a)\cup aX(a^{-1})=F=X(b)\cup bX(b^{-1}).}$ 

Banach–Tarski paradox edit

The most famous example of paradoxical sets is the Banach–Tarski paradox, which divides the sphere into paradoxical sets for the special orthogonal group. This result depends on the axiom of choice.

• Pathological (mathematics)