Identities of finite representations of groups in associative algebras (Q1803035)

Let \(A\) be an associative algebra over a finite field \(k\) and \(\varphi: G \to \text{Aut }A\) a fixed homomorphism of a group \(G\). Consider two alphabets \(X = \{x_ 1,x_ 2,\dots\}\), \(Y = \{y_ 1,y_ 2,\dots\}\) and the free associative algebra \(L^ X[Y]\) over \(k\) generated by the elements \(y^{f_ j}_ i\), where \(y_ i \in Y\), and \(f_ j\) is an element of the free group \(F_ \infty(X)\). An element \(u \in L^ X[Y]\) is called an identity of \((A,G)\) if for any pair of mappings \(\mu: X \to G\), \(\eta: Y \to A\) this element is transformed to zero (here \(a^ g\) is the result of the action of \(g\) on \(a\)). It is proved that there is a finite basis of identities in case when \((A,G)\) is finite, i.e. when \(A\) is generated by a finite set of elements \(a_ 1,\dots,a_ n\) together with \(a^ g_ i\) \(\forall g \in G\), \(i = 1,\dots,n\), and \(G/\text{Ker }\varphi\) is finite.