Matrotrophy associated with vertebrates and especially mammals, matrotrophy is found in 21 of 34 animal phyla, and is fairly common in 11 of those. It has arisen independently in more than 150 clades within Chordata and in more than 140 clades amongst invertebrates. Matrotrophy Matrotrophy is a form of maternal care during embryo development. Some flowering plants supplied the developing embryos for the first stages. The ovule has no stocks and the nutrients are provided by the "mother". In plants, matrotrophy is considered a critical evolutionary development preceding the origin of embryophytes and therefore essential to the evolution of land plants.
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