Let $G$ be a group containg more than 12 elements of order $13$; then which of the following is/are correct :
(A). $G$ is cyclic
(B). $G$ has a only one subgroup of order greater than $12$
(C). $G$ is never cyclic
(D). None of these
What of the following can we conclude about a group with more than $12$ elements of order $13$?
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$\begingroup$
abstract-algebra
group-theory
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1Suppose that $G$ was cyclic. What does that mean? Let $x$ be one of the elements of order $13$. What is $\langle x\rangle$, the subgroup generated by $x$, i.e. $\{1,x,x^2,x^3,\dots\}$. What is the order of $x^k$ for each $k\in\{1,\dots,12\}$? If there was another element of order $13$ other than these, where could it be? Does that make sense? Why not? – 2017-01-15
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0How to check group is cyclic or not.. – 2017-01-15
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0If you look up the definition, then that should be clear. A cyclic group is one that can be generated by a single element. – 2017-01-15
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0How to check group is cyclic .explain by formulas and theorems and what is lagrange's theorem? Describe in detail.. – 2017-01-15
1 Answers
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Hints:
1) A finite cyclic group has one, unique subgroup (also cyclic, of course) of any order dividing the group's order
2) A group of order a prime $\;p\;$ contains exactly $\;\phi(p)=p-1\;$ elements of order $\;p\;$