Endoskeleton three phyla and one subclass of animals, endoskeletons of various complexity are found: Chordata, Echinodermata, Porifera, and Coleoidea. An endoskeleton may function purely for support (as in the case of sponges), but often serves as an attachment site for muscle and a mechanism for transmitting muscular forces. A true endoskeleton is derived from mesodermal tissue. Such a skeleton is present in echinoderms and chordates. The poriferan 'skeleton' consists of microscopic calcareous or siliceous spicules or a spongin network. The Coleoidae do not have a true endoskeleton in the evolutionary sense; there, a mollusk exoskeleton evolved into several sorts of internal
Klein four-group representation on four points: In this representation, V is a normal subgroup of the alternating group A (and also the symmetric group S) on four letters. In fact, it is the kernel of a surjective group homomorphism from S to S. Other representations within S are: They are not normal subgroups of S According to Galois theory, the existence of the Klein four-group (and in particular, the permutation representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals, as established by Lodovico Ferrari: the map corresponds to the resolvent cubic,