Figure 3

Converting one character into another character using the cyclic group. (a) depicts the cyclic group Σ _c , which consists of the identity $s_0^{\prime }\equiv s_0$ and the three substitution types $s_1^{\prime },s_2^{\prime },s_3^{\prime }$ for nucleotide character states. (b) In order to convert the character GTAGA into ACCTC using this group, we need to introduce the operation $s^{\prime }:=(s_3^{\prime },s_1^{\prime },s_3^{\prime },s_1^{\prime },s_3^{\prime })$. As the operation s^′also transforms the constant character AAAAA to CGCGC, converting GTAGA into ACCTC is equivalent to evolving the character state A at the root along the tree such that the character CGCGC is attained at the leaves. The Fitch algorithm applied to the latter produces a unique most parsimonious solution of three substitutions as depicted by (c).