Ambulacraria Ambulacraria Ambulacraria or Coelomopora is a Superphylum of invertebrate phyla which includes echinoderms and hemichordates; a member of this group is called an ambulacrarian. Phylogenetic analysis suggest the echinoderms and hemichordates separated around 533 million years ago. The Ambulacraria are part of the deuterostomes, a larger clade that also includes the Chordata, Vetulicolia and Saccorhytus. The two living clades with representative organisms are: The group Xenoturbellida (two species of worm-like animals) has previously been considered to be in this clade, but is now considered to be placed more basally among Metazoans. Fossil taxa that may lie on the stem lineage:
Three subgroups lemma Three subgroups lemma In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of the Philip Hall and Ernst Witt's eponymous identity. In that which follows, the following notation will be employed: Let "X", "Y" and "Z" be subgroups of a group "G", and assume Then formula_4. More generally, if formula_5, then if formula_6 and formula_7, then formula_8. Hall–Witt identity If formula_9, then Proof of the three subgroups lemma Let formula_11, formula_12, and formula_13. Then formula_14, and by the Hall–Witt identity above, it follows that formula_15 and so formula_16. Therefore, formula_17