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517
help-10
ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:56:40
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:56:42
ok?
0
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:56:49
This is only possible since G is abelian
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:59:50
Yes
1
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:00:32
So now, what is the order of ab?
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:00:39
Remember gcd(a, b) = 1
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:03:50
It's mn
1
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:03:58
yes.
0
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:04:05
What does this tell you about G?
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1
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:04:53
The elements in G commute?
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:05:01
You already knew that. G is abelian.
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:05:13
However, you have just found an element in G which has order mn, the size of G.
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:07:21
|G| = mn. If G contains an element of order mn, that means it is **cyclic**
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:08:41
Oh
1
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:08:53
So now you know ab is a generator
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:09:00
G = <ab>
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:09:05
Sorry, I've been very distracted by everyone around me. Finally quiet🥺
1
0
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:09:08
is ok
0
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:09:25
https://cdn.discordapp.c…7622/unknown.png
unknown.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:09:55
So now we want an isomorphism between <ab> and <a> x <b>
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:10:00
make sense?
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:11:12
yes
1
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:11:42
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:11:46
Let's go with this example
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:11:52
when order of a = 2
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:11:54
order of b = 3
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:11:58
So mn = 6
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:12:08
So order of ab = 6 in G
0
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:12:19
We need to try to construct an isomorphism.
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:12:28
What seems something sensible if we want to succeed?
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:12:33
Just think about mapping 1 element for now
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:13:13
We're trying to show G is an isomorphism?
1
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:13:22
We want to **create** an isomorphism
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:13:28
between G and <a> x <b>
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:13:44
To do that we need to think of a mapping
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:13:57
Try thinking whats a good idea to try
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:14:39
things often turn out the way you expect them to in group theory
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:15:08
I'm unsure of what you mean by mapping
1
0
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:15:18
A function
0
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:15:23
might be referred to that or a map?
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:15:58
https://cdn.discordapp.c…1781/unknown.png
unknown.png
0
1
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:16:00
Because you have something like this
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:16:08
Elements from the domain 'map' to the codomain
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:16:15
1 is 'mapped' to 5 in my picture
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:16:27
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:16:50
For this, I am suggesting you think of a 'map' from elements of G to here
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:17:02
but don't try to map everything at once
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:17:07
Just 1 element at a time
0
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:17:31
So right now, maybe it's easier to map **from** G to <a> x <b>
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:17:46
There are actually only a few elements of G we know (explicitly)
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:17:58
So mapping from these elements to one another?
1
0
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:18:06
No, from group G
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:18:13
to this group <a> x <b>
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:18:21
What's an element in group G you know?
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:18:37
any
0
1
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null
517
help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:18:53
a
1
0
null
null
null
null
null
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:18:58
sure.
0
1
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:19:09
So what might be a good idea to map a to
0
1
321,445,119,171,756,000
Shuri2060
9525
a μ-er
false
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:19:12
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
null
null
null
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:19:13
in here
0
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:19:26
(a,e)?
1
0
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null
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:19:28
ok
0
1
null
null
null
null
null
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:19:32
so lets call our function f
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:19:36
f(a) = (a, e)
0
1
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:19:39
Another element from G?
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:19:49
b
1
0
null
null
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null
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:19:52
f(b) = ?
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:19:56
Just take a guess
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:20:00
if it doesn't work we can try again
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:20:00
(e,b)
1
0
null
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null
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null
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:20:03
ok
0
1
null
null
null
null
null
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:20:03
so
0
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:20:12
f(a) = (a, e) f(b) = (e, b)
0
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:20:22
Then it turns out this actually locks down our function
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:20:27
You chose where a and b map to
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:20:30
everything else must follow
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:20:42
if we want it to be an isomorphism
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:20:52
f(ab) = f(a)f(b) = (a, e)(e, b) = (a, b)
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:20:59
Good so far?
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:21:03
Mhm
1
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:21:06
Then you can also check
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:21:07
a^2
0
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:21:08
b^2
0
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:21:14
ab^2, etc.
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:21:22
I will tell you now what you chose works
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:21:31
But for the original question, we don't know if m and n are 2 and 3
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:21:37
So we need to prove this idea works in general
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:21:51
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:22:25
$$\langle a\rangle \times \langle b\rangle= \{(a, b):a\in \langle a\rangle, b\in \langle b\rangle\}$$
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 19:22:27
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:22:56
$$ord(a) = m$$ $$ord(b) = n$$
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 19:23:17
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:23:36
So try to do the same thing in this case, and in particular you need to prove what you construct is an isomorphism
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:24:36
That doesn't sound easy, but we can also use something we found our earlier
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:24:50
$ab$ generates G because its order is $mn$
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 19:24:52
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:25:49
And we want to show <ab> is isomorphism with <a>×<b>?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:25:55
yes exactly
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:26:54
Well, we got ab from that question, what about the other question it hints at? Where does this go into play?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:27:06
I think you used them already
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