Group with p elements of order 2

Groups, Rings, Domains, Modules, etc, Galois theory
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Grigorios Kostakos
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Group with p elements of order 2

#1

Post by Grigorios Kostakos »

Let \(G\) a non-commutative group of order \(2{\rm{p}}\), where \(\rm{p}\) is a prime number. Prove that \(G\) has \(\rm{p}\) discrete elements of order \(2\)
Grigorios Kostakos
achilleas
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Re: Group with p elements of order 2

#2

Post by achilleas »

Since \(G\) is non-abelian, \(p\) must be odd. Since \(p\) is a prime number that divides the order of \(G\), a theorem of Cauchy implies that \(G\) has an element of order \(p\), which generates a subgroup \(H\) of order \(p\). Since \(G\) is of order \(2p\), with \(p\) being odd, it has at most one subgroup of order \(p\). Thus every element of \(G\) not in \(H\) must be of order \(2\), that is, \(G\) has \(p\) elements of order \(2\).
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