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The groups are coming...

Source: RMO 2006

April 18, 2006
group theoryabstract algebrasuperior algebrasuperior algebra unsolved

Problem Statement

Let G\displaystyle G be a finite group of n\displaystyle n elements (n≥2)\displaystyle ( n \geq 2 ) and p\displaystyle p be the smallest prime factor of n\displaystyle n. If G\displaystyle G has only a subgroup H\displaystyle H with p\displaystyle p elements, then prove that H\displaystyle H is in the center of G\displaystyle G. Note. The center of G\displaystyle G is the set Z(G)={a∈G∣ax=xa, ∀x∈G}\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}.
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