Does this exist? I don't think it does. For any cyclic group the totient function of 10 is 4, so there is 4 of them. But also if one element is of order 10, say $a$, then $a^3$, $a^7$ is also of order 10 but thats three of them. Now I understand what I just did is mainly for cyclic groups, but can I apply that idea to non cyclic groups such as dihedrals or symmetric groups?
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asked Oct 7, 2014 at 12:24
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4$\begingroup$ You answered completely your own question. Check it carefully. +1 $\endgroup$– TimbucOct 7, 2014 at 12:29
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$\begingroup$ i wanted to make sure in because I havent really worked with dihedrals or symmetric groups that much. $\endgroup$– Jack ArmstrongOct 7, 2014 at 12:33
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$\begingroup$ what you know and mention about cyclic groups is more than enough since any group having an element of order ten has a cyclic subgroup of order ten... obviously. No dihedral or symmetric stuff required. $\endgroup$– TimbucOct 7, 2014 at 12:36
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$\begingroup$ It has nothing to do with dihedral or symmetric groups - see the answers. And moreover $\phi(10)\neq 2$. $\endgroup$– Dietrich BurdeOct 7, 2014 at 12:39
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2 Answers
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can I apply that idea to non cyclic groups such as dihedrals or symmetric groups?
Yes. If you have an element of order 10 in any group, it generates a subgroup of order 10, which is cyclic by definition. Since this subgroup has four generators, each of those is an element of order 10 in the subgroup, and is also of order 10 in the original group.
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$\begingroup$ What do you mean with "yes"? The OP asks if a group exists with exactly two elements of order 10 ... $\endgroup$ Oct 7, 2014 at 12:33
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$\begingroup$ There is only one question, which is "but can I apply that idea to non cyclic groups such as dihedrals or symmetric groups?". I answered that. $\endgroup$– fkraiemOct 7, 2014 at 12:34
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If $x$ is an element of order $10$, then also $x^3, x^7, x^9$ are all different elements of order 10. So it is impossible to have exactly two of them.
In general, if $n$ is a natural number and $G$ is a group with exactly $2$ elements of order $n$, then $n=3, 4$ or $6$. This follows from solving the equation $\varphi(n)=2$, where $\varphi$ is Euler's totient function. Each of the cases can be realized: take $G \cong D_n$, with $n=3,4$ or $6$.
Note that $D_3\cong S_3$ and $D_6\cong S_3 \times C_2$.
answered Oct 7, 2014 at 12:28