Publication Date:
2020
Description:
〈p〉Publication date: 1 June 2020〈/p〉
〈p〉〈b〉Source:〈/b〉 Journal of Algebra, Volume 551〈/p〉
〈p〉Author(s): Nathália Nogueira Gonçalves, Noraí Romeu Rocco〈/p〉
〈h5〉Abstract〈/h5〉
〈div〉〈p〉Let 〈em〉G〈/em〉 be a group and 〈em〉q〈/em〉 a non-negative integer. In this work we consider the 〈em〉q〈/em〉-tensor square 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉G〈/mi〉〈msup〉〈mrow〉〈mo〉⊗〈/mo〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mi〉G〈/mi〉〈/math〉 and the group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, a certain extension of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉G〈/mi〉〈msup〉〈mrow〉〈mo〉⊗〈/mo〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mi〉G〈/mi〉〈/math〉 by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉G〈/mi〉〈mo〉×〈/mo〉〈mi〉G〈/mi〉〈/math〉. Our interest is to study the behavior of these groups under the assumption that 〈em〉G〈/em〉 is a powerful finite 〈em〉p〈/em〉-group, 〈em〉p〈/em〉 a prime number. Under such assumptions we prove that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi mathvariant="normal"〉exp〈/mi〉〈mo〉〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 divides 〈em〉q〈/em〉, then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉G〈/mi〉〈msup〉〈mrow〉〈mo〉⊗〈/mo〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mi〉G〈/mi〉〈/math〉 is also powerful and, additionally, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈msup〉〈mrow〉〈mo〉⊗〈/mo〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉d〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mi〉d〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 denotes the minimal number of generators of 〈em〉G〈/em〉. We also establish bounds for the exponent of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉G〈/mi〉〈msup〉〈mrow〉〈mo〉⊗〈/mo〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mi〉G〈/mi〉〈/math〉 in terms of the exponent of 〈em〉G〈/em〉. We derive our results via the embedding of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉G〈/mi〉〈msup〉〈mrow〉〈mo〉⊗〈/mo〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mi〉G〈/mi〉〈/math〉 into 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. To this end we prove that all terms of the lower central series and of the derived series of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 are powerfully embedded in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, with the only exception of the whole group itself. We give a simple example to show that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is not necessarily powerful. Our results extend to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈mi〉q〈/mi〉〈mo〉≥〈/mo〉〈mn〉0〈/mn〉〈/math〉 similar results found by Moravec [11] for the non-abelian tensor square 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈mi〉G〈/mi〉〈mo〉⊗〈/mo〉〈mi〉G〈/mi〉〈/math〉 in the case 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉.〈/p〉〈/div〉
Print ISSN:
0021-8693
Electronic ISSN:
1090-266X
Topics:
Mathematics
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