Prove-that-a-group-G-of-prime-order-is-cyclic-

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 \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 .