Annali di Matematica Pura ed Applicata ( IF 1 ) Pub Date : 2020-07-22 , DOI: 10.1007/s10231-020-01020-2 Eloisa Detomi , Marta Morigi , Pavel Shumyatsky
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:
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\(2 ^ {\ aleph _0} \)。