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Amalgams of finite groups provide another example. Let $A$ and $B$ be finite groups and let $C = A \cap B.$ Suppose that $P$ is a Sylow $p$-subgroup of $A$, and that $C$ contains a Sylow $p$-subgroup of $B.$ Then the amalgam $A*_{C}B$ has a unique conjugacy class of maximal finite $p$-subgroups, but is an infinite group as long as $C$ is proper in both $A$ and $B.$. In fact, the process an then iterated to the case where $A$ and $B$ may themselves be amalgams of finite groups of this type, and so on. For general results on amalgams, see J-P. Serre's book "Trees". For applications of this type of construction to fusion systems on finite $p$-groups, see two recent papers of mine in Journal of Algebra and Transactions of the AMS.