Factor complexity C and palindromic complexity P of
infinite words with language closed under reversal are known to be related by
the inequality P(n)+P(n+1)≤2+C(n+1)−C(n) for any n∈N\,. Word for which
the equality is attained for any n is usually called rich in palindromes. In
this article we study words whose languages are invariant under a finite group
G of symmetries. For such words we prove a stronger version of the above
inequality. We introduce notion of G-palindromic richness and give several
examples of G-rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur