The component Dyer–Lashof coalgebras as subcoalgebras of free unstable coalgebras

Let R[n]R[n] be the coalgebra of length n   homology operations on infinite loop spaces. We compare R[n]R[n] with a cofree unstable coalgebra using its hom-dual image which is isomorphic to a subalgebra, SEDnSEDn, of the extended Dickson algebra EDnEDn. SEDnSEDn is isomorphic to a subalgebra of a free unstable algebra on two generators. This result is related to the Peterson conjecture proven by Pengelley and Williams for the odd prime version of the classical Dickson algebra DnDn.