An abelian square is a word 
in which 
and 
are nonempty finite words and each
letter occurs equally often in 
and 
. Equivalently, 
is a permutation of the letters
of 
. For example, 
is an abelian square, although its two halves are different.
An ordinary square 
is always an abelian square.
There is an infinite word over a four-letter alphabet containing no abelian square. Three letters do not suffice, since every three-letter word of length at least eight contains one (Salo et al. 2026). The analogous continuous abelian square problem has a different answer.
