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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:33:40
C2
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:33:41
is just
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321,445,119,171,756,000
Shuri2060
02/05/2022 18:33:43
{e, a}
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321,445,119,171,756,000
Shuri2060
02/05/2022 18:34:49
All good?
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589,206,850,839,380,000
Moon Child
02/05/2022 18:35:01
Ah yes
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Shuri2060
02/05/2022 18:35:09
So let's stick with this example
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:35:15
So in the original question let's say m = 6
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:35:31
So I now have <a> x < b>
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:35:36
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:35:50
a = 2, b = 3
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:35:54
and they are relatively prime.
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:36:12
You now need to show this group must be isomorphic to the original group
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:36:29
I'm still confused how you get the whole thing
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:36:33
ok
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:36:39
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:36:48
You understand I got this from the original definition, yes or no?
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ee650a1eb2184f5ca1221029aa080be4
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Moon Child
02/05/2022 18:37:27
Well I understand {e,a} in C2 and {e,a,b} in C3?
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:37:31
$$C_2 = \{e, a\}$$ $$C_3 = \{e, b, b^2\}$$
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ee650a1eb2184f5ca1221029aa080be4
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Moon Child
02/05/2022 18:37:39
Or is it e b, and b2
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help-10
ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:37:44
**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 18:37:47
Oh
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:37:49
C_n is the cyclic group of order n
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:37:53
There is only 1 generator
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:37:55
for cyclic groups
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:38:17
You are familiar with that right?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:38:53
A finite group of size n is cyclic if one element has order n. If that is true, then all the elements must be some power of that element
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:38:59
Then everything must be generated by it.
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ee650a1eb2184f5ca1221029aa080be4
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Moon Child
02/05/2022 18:39:10
Oh okay
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:39:45
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:39:48
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:39:56
So now this comes from me just writing out all the possible elements
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Moon Child
02/05/2022 18:40:13
Mhm
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:40:26
So to familiarize yourself with this
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:40:33
What is (a, b)^2
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:41:15
Written in this way? Or?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:41:25
You can go step by step
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:41:28
So first
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:41:36
(a, b)^2 = (a, b)(a, b)
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:41:59
(a+a,b+b)?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:42:14
That all depends on the group operation
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:42:20
but we usually prefer to write like multiplication
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:42:29
$(a, b)^2 = (a, b)(a, b) = (a^2, b^2)$
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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 18:42:31
So like this.
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:42:38
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:43:05
For the general case it will be written like this (the multiplication sign is omitted, but they **are** different)
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ee650a1eb2184f5ca1221029aa080be4
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Moon Child
02/05/2022 18:43:19
Oh okay
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:43:27
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:43:33
But then, this must correspond to one of these
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Shuri2060
02/05/2022 18:43:34
which?
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ee650a1eb2184f5ca1221029aa080be4
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Moon Child
02/05/2022 18:44:01
(e,b^2)
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:44:06
right
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:44:10
because a^2 = e
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ee650a1eb2184f5ca1221029aa080be4
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Moon Child
02/05/2022 18:44:15
Mhn
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:44:26
For clarity, sometimes it might be written like this
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:44:35
$$\{(e_A, e_B), (a, e_B), (e_A, b), (a, b), (e_A, b^2), (a, b^2)\}$$
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help-10
ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:44:38
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:44:45
Because e_A and e_B are not necessarily the same
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:44:52
They are just the identity elements from A and B
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:45:07
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:45:23
This is used when it is clear from context or doesn't matter.
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:45:36
So which of these is the identity element of A x B ?
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:46:26
(e,e)?
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:46:34
yep
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:46:40
I think you get how this group works
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:46:48
I do now
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:46:50
https://cdn.discordapp.c…6210/unknown.png
unknown.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:46:54
So the question...
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:47:08
You need to show this thing is isomorphic to G
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:47:34
Do you remember how an isomorphism works, roughly?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:47:38
Just the main concept
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:48:09
Show it's bijective and morphic
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:48:16
yah but the main concept is
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:48:19
If we have 2 groups
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:48:21
X and Y
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:48:24
and they are isomorphic
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:48:36
I can 'relabel' the things in X with some things in Y
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:48:47
And the group will still behave exactly the same
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:48:57
Just different labels (and group operation)
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:49:17
f(xy)=f(x)f(y)
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:49:29
Yes, that is homomorphism. And then you also need bijection
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:50:15
So this isn't an easy question, there are several steps to it
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:50:26
You could focus on group G first.
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:50:37
It tells you $\gcd(m, n) = 1$
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:50:39
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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1
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:50:50
a has order m b has order n
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:51:23
Hmm
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:52:00
Thinking 😂
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:53:06
Do you have any idea what this could tell you about G?
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:54:31
I'm no sure
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:54:57
So we have G is abelian also, forgot to mention
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:55:05
But think about the element $ab$
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:55:07
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:55:56
If this helps, these are the problems that the question hints about
https://cdn.discordapp.c…20205_155537.jpg
IMG_20220205_155537.jpg
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:56:06
Right exactly
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:56:07
$(ab)^2 = (ab)(ab) = a^2b^2$
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help-10
ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:56:09
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:56:15
First of all, the group is abelian
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:56:19
So when we take powers of ab
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:56:23
We can rearrange like this
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:56:37
$$(ab)^k = a^kb^k$$
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