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| ==Statement== | | ==Statement== |
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| Suppose <math>G</math> is a group such that the [[fact about::cube map;2| ]][[cube map]] <math>x \mapsto x^3</math> is an [[endomorphism]] of <math>G</math>. Further, suppose that <math>G</matH> is 2-divisible, i.e., every element of <math>G</math> is a [[square element]]. | | Suppose <math>G</math> is a group such that the [[fact about::cube map;2| ]][[cube map]] <math>x \mapsto x^3</math> is an [[endomorphism]] of <math>G</math>. |
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| Then, <math>G</math> is a [[nilpotent group]] and its [[nilpotency class]] is at most four. It is not yet clear whether the nilpotency class should be at most three. | | Then, <math>G</math> is a [[nilpotent group]] and its [[nilpotency class]] is at most three. |
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| Note that the condition of 2-divisibility is true for any [[odd-order group]], and more generally, any group in which every element has finite odd order. It is also true for any [[rationally powered group]] and in fact for any group powered over the prime 2.
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| ==Related facts==
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| * [[Cube map is surjective endomorphism implies abelian]]
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| * [[Cube map is endomorphism iff abelian (if order is not a multiple of 3)]]
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| ==Facts used== | | ==Facts used== |
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| # [[uses::nth power map is endomorphism implies every nth power and (n-1)th power commute]] | | # [[uses::Levi's characterization of 3-abelian groups]] |
| # [[uses::Exponent three implies class three for groups]] | | # [[uses::2-Engel implies class three for groups]] |
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| ==Proof== | | ==Proof== |
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| '''Given''': A group <math>G</math> such that the map <math>\sigma = x \mapsto x^3</math> is an endomorphism of <math>G</math>. Further, every element of <math>G</math> has finite odd order.
| | The proof follows directly by combining Facts (1) and (2). |
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| '''To prove''': <math>G</math> is nilpotent and its nilpotency class is at most four.
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| '''Proof''':
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| {| class="sortable" border="1"
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| ! Fact no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
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| | 1 || <math>\sigma(G)</math> is a characteristic subgroup of <math>G</math> and <math>G/\sigma(G)</math>, if nontrivial, is a group of exponent three || || <math>\sigma = x \mapsto x^3</math> is an endomorphism|| || {{fillin}} -- direct reasoning
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| | 2 || <math>\sigma(G)</math> is in the center of <math>G</math> || Fact (1) || <math>\sigma = x \mapsto x^3</math> is an endomorphism<br><math>G</math> is 2-divisible|| || By Fact (1), every cube commutes with every square. Thus, every element of <math>\sigma(G)</math> commutes with every square. By 2-divisibility, every element of <math>G</matH> is a square, so we obtain that every element of <math>\sigma(G)</math> commutes with every element of <math>G</math>. Thus, <math>\sigma(G)</math> is in the center of <math>G</math>.
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| | 3 || <math>G/\sigma(G)</math> is a nilpotent group of nilpotency class at most three. || Fact (2) || || Step (1) || Step-fact combination direct
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| | 4 || <math>G</math> is a nilpotent group of nilpotency class at most four. || || || Steps (2), (3) || Step-combination direct, plus the observation that if the quotient by a subgroup in the center has class at most <math>c</math>, then the group itself has class at most <math>c+1</math>.
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| |}
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