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Let be a group and be an integer greater than or equal to . is said to be -permutable if every product of elements can be reordered at least in one way. We prove that, if has a centre of finite index , then is -permutable. More bounds are given on the least such that is -permutable.
Let G be a supersolvable finite group or a solvable one, in which the orders of principal factors are primes or squares of primes; and let exp G divide ( prime numbers). A much simple enunciation is given for essentially known theorems on some systems of laws characterizing G; it is indeed shown that the laws 1.(1) and 1.(2) can be removed from the enunciates of such theorems, getting equivalent systems of laws characterizing G.
Let be a group and be an integer greater than or equal to . is said to be -permutable if every product of elements can be reordered at least in one way. We prove that, if has a centre of finite index , then is -permutable. More bounds are given on the least such that is -permutable.
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