Answer to Question #144684 in Abstract Algebra for J

Question #144684
Assume that (G,*) is a group, and that |G|=p, where p is prime. Prove that G is cyclic.
1
Expert's answer
2020-11-17T07:39:33-0500

Let that "(G,*)" be a group of order"|G|=p" , where "p" is prime. Let "a\\in G" and "a\\ne e" where "e" is the identity element of "(G,*)". According to Lagrange's theorem, the order of cyclic subgroup "\\langle a\\rangle" divides "|G|=p". Therefore, "|\\langle a\\rangle|\\in\\{1,p\\}". Taking into account that the identity "e" is a unique element of order 1, we conclude that "|\\langle a\\rangle|>1", and therefore, "|\\langle a\\rangle|=p". Since "\\langle a\\rangle\\subseteq G" and "|\\langle a\\rangle|=p=|G|", we conclude that "\\langle a\\rangle=G." Consequently, G is a cyclic group generated by "a".


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS