Exponents and the cohomology of finite groups

Abstract

We will provide an example of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which has elements of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p cubed"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">p^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in some of its integral cohomology groups but which also has the property that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p squared"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">p^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> annihilates <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H With bar Superscript i Baseline left-parenthesis upper G semicolon double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo stretchy="false"> ¯ </mml:mo> </mml:mover> </mml:mrow> <mml:mi>i</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar {H}^i(G;\mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all sufficiently high <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i"> <mml:semantics> <mml:mi>i</mml:mi> <mml:annotation encoding="application/x-tex">i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This provides a counterexample to a conjecture of A. Adem which states that if a finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has an element of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Superscript n"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">p^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in one of its integral cohomology groups, then it has such an element in infinitely many of its cohomology groups.