For a group G and a finite set A, denote by \(\mathrm{End}(A^G)\) the monoid of all continuous shift commuting self-maps of \(A^G\) and by \(\mathrm{Aut}(A^G)\) its group of units. We study the minimal cardinality of a generating set, known as the rank, of \(\mathrm{End}(A^G)\) and \(\mathrm{Aut}(A^G)\). In the first part, when G is a finite group, we give upper and lower bounds for the rank of \(\mathrm{Aut}(A^G)\) in terms of the number of conjugacy classes of subgroups of G. In the second part, we apply our bounds to show that if G has an infinite descending chain of normal subgroups of finite index, then \(\mathrm{End}(A^G)\) is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups.

为一组G ^和一组有限的甲，表示由\（\ mathrm {完}（A ^ G）\）的所有连续移位的通勤的自映射的幺\（A ^ g \）和由\（\ mathrm {Aut}（A ^ G）\）其单位组。我们研究了发电机组的最小基数，被称为秩，的\（\ mathrm {完}（A ^ G）\）和\（\ mathrm {AUT}（A ^ G）\） 。在第一部分中，当G ^是一个有限群，我们给出的等级的上限和下限\（\ mathrm {AUT}（A ^ G）\）中的亚组的共轭类的数量方面ģ。在第二部分中，我们应用边界来表明如果G具有有限索引的普通子组的无限下降链，则\（\ mathrm {End}（A ^ G）\）并不是有限生成的；广泛类别的无限组就是这种情况，例如无限残差有限或无限局部分级的组。