Prove that a group G of prime order is cyclic.


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Question Number 193339 by Rajpurohith last updated on 10/Jun/23

$P r o v e t h a t a g r o u p G o f p r i m e o r d e r i s c y c l i c .$
	Answered by witcher3 last updated on 10/Jun/23
	$let g∈G−e,existe since ord(G)=p≥2 g^p =e G contien at lest 2 element let z→^f G k→g^k morphisme of Group ker f={k∈Z suche g^k =e} ⇒p∣k⇒k=pm use Theorem of morphisme f induce isomorphisme of Z↙pZ→g ⇒g∼Z↙pZ⇒g Cyclic$
	$let g \in G - e, existe since ord (G) = p ⩾ 2$ $g^{p} = e$ $G contien at lest 2 element$ $let z \overset{f}{\to} G$ $k \to g^{k}$ $morphisme of Group$ $\ker f = {k \in Z suche g^{k} = e}$ $\Rightarrow p ∣ k \Rightarrow k = pm$ $use Theorem of morphisme f induce isomorphisme$ $of Z ↙ pZ \to g$ $\Rightarrow g \sim Z ↙ pZ \Rightarrow g Cyclic$
	Commented by Rajpurohith last updated on 11/Jun/23

	$V e r y n i c e s i r, T h a n k s .$
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