An integral version of a classical embedding theorem concerning quaternion algebras B over a number field k is proved. Assume that B satisfies the Eichler condition, that is, some infinite place of k is not ramified in B, and let Ω be an order in a quadratic extension of k. The maximal orders of B which admit an embedding of Ω are determined. Although most Ω embed into either all or none of the maximal orders of B, it turns out that some Ω are ' selective ', in the sense that they embed into exactly half of the isomorphism types of maximal orders of B. As an application, the maximal arithmetic subgroups of B*\k* which contain a given element of B*\k* are determined.

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In this paper we determine the universal deformation rings of certain modular representations of finite groups which belong to cyclic blocks. The representations we consider are those for which every endomorphism is stably equivalent to multiplication by a scalar. We then apply our results to study the counterparts for universal deformation rings of conjectures about embedding problems in Galois theory.

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