**Symmetries of Group Actions**

Let *G* be a group and *X* a topological space. Any action of *G* on *X* gives a homomorphism, where *F(X)* is an appropriate algebra of *k*-valued functions, such as the Gelfand-Naimark algebra of continuous functions vanishing at infinity. is defined by with the adjoint defined by

for, and .

This may be described by a linear mapping

where, are the elements of *G*, and, which has the property that group-like elements in *kG* give rise to automorphisms of *F(X)*.

endows *F(X)* with an important extra structure, described below.

Read more about this topic: Group Hopf Algebra

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—Tacitus (c. 55–117)

“We begin with friendships, and all our youth is a reconnoitering and recruiting of the holy fraternity they shall combine for the salvation of men. But so the remoter stars seem a nebula of united light, yet there is no *group* which a telescope will not resolve; and the dearest friends are separated by impassable gulfs.”

—Ralph Waldo Emerson (1803–1882)