Counting group elements of order 𝑝 modulo 𝑝²

Marcel Herzog

Type: Article

Publication Date: 1977-01-01

Citations: 5

DOI: https://doi.org/10.1090/s0002-9939-1977-0466316-5

Abstract

Let <italic>G</italic> be a finite group of order divisible by the prime <italic>p</italic>. It is shown that the number of elements of <italic>G</italic> of order <italic>p</italic> is congruent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 1"> <mml:semantics> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">- 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> modulo <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{p^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, unless a Sylow <italic>p</italic>-subgroup of <italic>G</italic> is cyclic, generalized quaternion, dihedral or quasidihedral.

Locations

Proceedings of the American Mathematical Society - View - PDF

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