Calculating the order of two function in some group
$begingroup$
While studying a subject about the group operator I came across with the following question:
Consider the following function: $h_1,h_2,:,mathbb{R}tomathbb{R}$
where $h_1=-x$ and $h_2=-x+1$. Lets define a new group $X=langle h_1,
h_2rangle$ which follows $Xcurvearrowrightmathbb{R}$ by $h_1cdot x
= h_1(x)$.
Calculate $o(h_1)$ and $o(h_2)$.
Unfortunately, the books does not have final answer or solutions. I started from using the following definition of order: Let $G$ be a group, the order of $gin G$, meaning $o(g)$ is the minimal $nin mathbb{N}$ so $g^n=e$.
By using this definition, I understand that I need to find the minimal $n,minmathbb{N}$ so $h_1^n=e$ and $h_2^m=e$. But I'm not sure how to use the that $Xcurvearrowright mathbb{R}$.
group-theory
asked Dec 22 '18 at 15:42
vesiivesii
3978
$endgroup$
add a comment |
$begingroup$
While studying a subject about the group operator I came across with the following question:
Consider the following function: $h_1,h_2,:,mathbb{R}tomathbb{R}$
where $h_1=-x$ and $h_2=-x+1$. Lets define a new group $X=langle h_1,
h_2rangle$ which follows $Xcurvearrowrightmathbb{R}$ by $h_1cdot x
= h_1(x)$.
Calculate $o(h_1)$ and $o(h_2)$.
Unfortunately, the books does not have final answer or solutions. I started from using the following definition of order: Let $G$ be a group, the order of $gin G$, meaning $o(g)$ is the minimal $nin mathbb{N}$ so $g^n=e$.
By using this definition, I understand that I need to find the minimal $n,minmathbb{N}$ so $h_1^n=e$ and $h_2^m=e$. But I'm not sure how to use the that $Xcurvearrowright mathbb{R}$.
group-theory
asked Dec 22 '18 at 15:42
vesiivesii
3978
$endgroup$
$begingroup$
What books are you using?
$endgroup$
– Shaun
Dec 22 '18 at 15:52
add a comment |
$begingroup$
While studying a subject about the group operator I came across with the following question:
Consider the following function: $h_1,h_2,:,mathbb{R}tomathbb{R}$
where $h_1=-x$ and $h_2=-x+1$. Lets define a new group $X=langle h_1,
h_2rangle$ which follows $Xcurvearrowrightmathbb{R}$ by $h_1cdot x
= h_1(x)$.
Calculate $o(h_1)$ and $o(h_2)$.
Unfortunately, the books does not have final answer or solutions. I started from using the following definition of order: Let $G$ be a group, the order of $gin G$, meaning $o(g)$ is the minimal $nin mathbb{N}$ so $g^n=e$.
By using this definition, I understand that I need to find the minimal $n,minmathbb{N}$ so $h_1^n=e$ and $h_2^m=e$. But I'm not sure how to use the that $Xcurvearrowright mathbb{R}$.
group-theory
asked Dec 22 '18 at 15:42
vesiivesii
3978
$endgroup$
While studying a subject about the group operator I came across with the following question:
Consider the following function: $h_1,h_2,:,mathbb{R}tomathbb{R}$
where $h_1=-x$ and $h_2=-x+1$. Lets define a new group $X=langle h_1,
h_2rangle$ which follows $Xcurvearrowrightmathbb{R}$ by $h_1cdot x
= h_1(x)$.
Calculate $o(h_1)$ and $o(h_2)$.
Unfortunately, the books does not have final answer or solutions. I started from using the following definition of order: Let $G$ be a group, the order of $gin G$, meaning $o(g)$ is the minimal $nin mathbb{N}$ so $g^n=e$.
By using this definition, I understand that I need to find the minimal $n,minmathbb{N}$ so $h_1^n=e$ and $h_2^m=e$. But I'm not sure how to use the that $Xcurvearrowright mathbb{R}$.
group-theory
group-theory
asked Dec 22 '18 at 15:42
vesiivesii
3978
asked Dec 22 '18 at 15:42
vesiivesii
3978
asked Dec 22 '18 at 15:42
vesiivesii
3978
asked Dec 22 '18 at 15:42
vesiivesii
3978
asked Dec 22 '18 at 15:42
vesiivesii
3978
3978
$begingroup$
What books are you using?
$endgroup$
– Shaun
Dec 22 '18 at 15:52
add a comment |
$begingroup$
What books are you using?
$endgroup$
– Shaun
Dec 22 '18 at 15:52
$begingroup$
What books are you using?
$endgroup$
– Shaun
Dec 22 '18 at 15:52
$begingroup$
What books are you using?
$endgroup$
– Shaun
Dec 22 '18 at 15:52
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint/Explanation:
You should use the group action axioms, in particular this one:
If $g,hin G$ elements and $G$ acts on $X$ then $$g.(h.x) = (gh).x$$
Using this you should calculate $h_1^2.x$ and $h_2^2.x$
answered Dec 22 '18 at 15:55
YankoYanko
8,4692830
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint/Explanation:
You should use the group action axioms, in particular this one:
If $g,hin G$ elements and $G$ acts on $X$ then $$g.(h.x) = (gh).x$$
Using this you should calculate $h_1^2.x$ and $h_2^2.x$
answered Dec 22 '18 at 15:55
YankoYanko
8,4692830
$endgroup$
add a comment |
$begingroup$
Hint/Explanation:
You should use the group action axioms, in particular this one:
If $g,hin G$ elements and $G$ acts on $X$ then $$g.(h.x) = (gh).x$$
Using this you should calculate $h_1^2.x$ and $h_2^2.x$
answered Dec 22 '18 at 15:55
YankoYanko
8,4692830
$endgroup$
add a comment |
$begingroup$
Hint/Explanation:
You should use the group action axioms, in particular this one:
If $g,hin G$ elements and $G$ acts on $X$ then $$g.(h.x) = (gh).x$$
Using this you should calculate $h_1^2.x$ and $h_2^2.x$
answered Dec 22 '18 at 15:55
YankoYanko
8,4692830
$endgroup$
Hint/Explanation:
You should use the group action axioms, in particular this one:
If $g,hin G$ elements and $G$ acts on $X$ then $$g.(h.x) = (gh).x$$
Using this you should calculate $h_1^2.x$ and $h_2^2.x$
answered Dec 22 '18 at 15:55
YankoYanko
8,4692830
answered Dec 22 '18 at 15:55
YankoYanko
8,4692830
answered Dec 22 '18 at 15:55
YankoYanko
8,4692830
answered Dec 22 '18 at 15:55
YankoYanko
8,4692830
8,4692830
add a comment |
add a comment |
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$begingroup$
What books are you using?
$endgroup$
– Shaun
Dec 22 '18 at 15:52