A coprime commutator in a profinite group G is an element of the form [x, y], where x and y have coprime order and an anti-coprime commutator is a commutator [x, y] such that the orders of x and y are divisible by the same primes. In the present paper, we establish that a profinite group G is finite-by-pronilpotent if the cardinality of the set of coprime commutators in G is less than \(2^{\aleph _0}\). Moreover, a profinite group G has finite commutator subgroup \(G'\) if the cardinality of the set of anti-coprime commutators in G is less than \(2^{\aleph _0}\).

中文翻译：

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互质和反互质换向器的简洁性

有限群G中的互质换向子是形式为[ x，  y ]的元素，其中x和y具有互质阶数，反互质换向子是换向子[ x，  y ]，使得x和y的阶数为可被相同的素数整除。在本文中，如果在G中的互质交换子集合的基数小于\（2 ^ {\ aleph _0} \），则我们确定一个有限群G是乘乘有限的。此外，有限群G具有有限的换向子群\（G'\）如果G中的一组反互质换向子的基数小于\（2 ^ {\ aleph _0} \）。