Let R be an algebra over a field and G a finite group of automorphisms and anti-automorphisms of R. We prove that if R satisfies an essential G-polynomial identity of degree d, then the G-codimensions of R are exponentially bounded and R satisfies a polynomial identity whose degree is bounded by an explicit function of d. As a consequence we show that if R is an algebra with involution * satisfying a * -polynomial identity of degree d, then the * -codimensions of R are exponentially bounded;

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... s gives a new proof of a theorem of Amitsur stating that in this case R must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.