We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [9], automorphism groups of quandles (up to isomorphisms) of order less than or equal to 5 were given. With the help of the software Maple, we compute the inner and automorphism groups of all seventy three quandles of order six listed in the appendix of [4]. Since computations of automorphisms of quandles relates to the problem of classification of quandles, we also describe an algorithm implemented in C for computing all quandles (up to isomorphism) of order less than or equal to nine.

a b s t r a c tOur aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra k [[G]] of a profinite group G, where k is a finite field of characteristic p.We define the concept of relative projectivity for a profinite k[[G]]-module. We prove a characterization of finitely generated relatively projective modules analogous to the finite case with additions of interest to the profinite theory. We introduce vertices and sources for indecomposable finitely generated k [[G]]-modules and show that the expected conjugacy properties hold-for sources this requires additional assumptions. Finally we prove a direct analogue of Green's Indecomposability Theorem for finitely generated modules over a virtually pro-p group.

We consider an intermediate category between the category of finite quivers and a certain category of pseudocompact associative algebras whose objects include all pointed finite dimensional algebras. We define the completed path algebra and the Gabriel quiver as functors. We give an explicit quotient of the category of algebras on which these functors form an adjoint pair. We show that these functors respect ideals, obtaining in this way an equivalence between related categories.

2 Definitions, terminology and background 2.1 Algebras and modules Our coefficient ring R will always be a commutative pseudocompact ring. In later sections we will require further structure on R, the main coefficient rings of interest to us being complete discrete valuation rings.Let Λ be a pseudocompact R-algebra (we follow the treatments in [9] and [18]). Examples of particular interest are the completed group algebra R[[G]] of a profinite group G, or later the group algebra RG of a finite group G. We consider the following categories of modules for Λ:• Λ-Mod C : the category whose objects are pseudocompact left Λ-modules [9, §1]. Objects of this category are complete, Hausdorff topological Λ-modules U having a basis of open neighbourhoods of 0 consisting of submodules V for which U/V has finite length. In other words, the category of inverse limits of left Λ-modules of finite length over R.• Λ-Mod D : the category whose objects are those topological Λ-modules having the discrete topology. Such modules are precisely the direct limits of left Λ-modules of finite length over R. We will call such modules discrete (they are called "locally finite" in [18]).• Λ-Mod abs : the category of abstract left Λ-modules Proof. We mimic the proof of [38, Lem. 52.2]. Write X = I X i with each X i an RG-sublattice of X of R-rank 1. The sequence remains split over R and the X i are indecomposable R-modules, so by Proposition 3.5 applied to these R-modules there is a subset J of I such that X = s(X ′ ) ⊕ J X i , and this remains true over RG. Thus the RG-module homomorphism h ′ = h| J Xi is an isomorphism, where h is the map induced by h. By Nakayama's Lemma, h| J Xi = h ′ : J X i → X ′′ is an isomorphism. Now the composition of h ′−1 with the inclusion J X i → X is a splitting of h.

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