(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let H be a normal subgroup of prime order p in a finite group G. Suppose that p is the smallest prime dividing |G|. Prove that H is in the center Z(G).

2. Relevant equations

the Class Equation?

Sylow theorems are in the next section, so presumably this is to be done without them.

3. The attempt at a solution

Not completely sure of a solution, but here's (at least some of) what we know:

1. Since H is normal, [tex]ghg^{-1} \in H[/tex].

2. Since [tex]|H|[/tex] is prime, [tex]H[/tex] is cyclic and abelian.

3. [tex]G[/tex] is finite, with order [tex]|G| = p^nq[/tex].

4. The normalizer [tex]N(H)[/tex] (stabilizer under conjugation) is all of [tex]G[/tex]...

5. ...so [tex]|G| = |N(H)|[/tex] ??

6. Probably some more relevant properties.

And we want to show that [tex]H \subseteq Z(G)[/tex], i.e. [tex]H \subseteq \{g \in G | gx = xg \forall x \in G\}[/tex]

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# Homework Help: Normal subgroup of prime order in the center

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