An algebra is bicommutative if it satisfies left and right symmetries; i.e., \(a(bc)=b(ac)\) and \((ab)c=(ac)b\) . Let K be a field of characteristic zero, and \(M_n\) , \(n\ge 3\) , be the free metabelian bicommutative algebra generated by a set \(X_n=\{x_1,\ldots ,x_n\}\) of variables, in which the identity \((xy)(zt)=0\) is being satisfied. We define the action of the alternating group \(A_n\) on \(M_n\) as follows. \(\pi f(x_1,\ldots ,x_n)=f(x_{\pi (1)},\ldots ,x_{\pi (n)})\) , where \(\pi \in A_n\) and \(f\in M_n\) . The set \(M_n^{A_n}=\{f\in M_n\mid \pi f=f\ , \forall \pi \in A_n\}\) is a subalgebra of \(M_n\) called the algebra of invariants of the group \(A_n\) . In the first part of this study, we describe the elements of the algebra \(M_n^{A_n}\) . We also give the description of the algebras \(M_2^{C_2}\) , \(M_2^{C_3}\) , \(M_2^{C_2\times C_2}\) , and \(M_2^{C_4}\) of invariants of the groups \(C_2\) , \(C_3\) , \(C_2\times C_2\) , and \(C_4\) of order up to 4, respectively, as a subgroups of the general linear group \(\text {GL}_2(K)\) .
