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help-10
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M8732
02/05/2022 18:03:13
How do you define isomorphic then?
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Mohmoka
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Shuri2060
02/05/2022 18:03:15
Do you get the Q?
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321,445,119,171,756,000
Shuri2060
02/05/2022 18:03:32
The idea behind it
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Moon Child
02/05/2022 18:03:52
A slight idea about it but I'm not too sure
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Shuri2060
02/05/2022 18:04:09
I usually find it helpful to use small examples
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Moon Child
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Moon Child
false
517
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Shuri2060
02/05/2022 18:04:13
Say C_3 and C_5
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:04:30
Then what does the direct product look like
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help-10
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Shuri2060
02/05/2022 18:04:45
Just to get you thinking in the right direction of what the question is asking
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help-10
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Shuri2060
02/05/2022 18:05:10
I think the hard bit is seeing what the group <a> x <b> looks like if you're unfamiliar with direct products
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Shuri2060
02/05/2022 18:07:00
@Moon Child Sorry that was a bit badly written by me 😂
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Moon Child
02/05/2022 18:07:12
👀
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Moon Child
02/05/2022 18:07:34
C_5 and C_3 as in the cyclic groups?
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Shuri2060
02/05/2022 18:08:38
yh well...
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Shuri2060
02/05/2022 18:08:43
Basically yes
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:08:47
My example is
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help-10
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Shuri2060
02/05/2022 18:08:49
m = 15
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Shuri2060
02/05/2022 18:08:57
a has order 3 b has order 5
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Moon Child
9119
Moon Child
false
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Shuri2060
02/05/2022 18:09:03
so <a> is the same as C_3
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help-10
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Shuri2060
02/05/2022 18:09:09
<b> is the same as C_5
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help-10
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Shuri2060
02/05/2022 18:09:16
if that makes sense? same structure/isomorphic
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Moon Child
02/05/2022 18:10:36
I'm a bit confused about cyclic groups
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ee650a1eb2184f5ca1221029aa080be4
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Moon Child
02/05/2022 18:10:44
I have a slight idea but not well explained
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Shuri2060
02/05/2022 18:11:21
Alright let me see
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Shuri2060
02/05/2022 18:11:29
$\{e, a, a^2\}$
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TeXit
02/05/2022 18:11:32
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:11:35
This is what C_3 looks like
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:11:38
in general
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Shuri2060
02/05/2022 18:11:43
The structure will always be the same
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Shuri2060
02/05/2022 18:12:02
$C_n = \{e, a, a^2, \cdots, a^{n-1}\}$
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TeXit
02/05/2022 18:12:04
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:12:10
You get this part?
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Shuri2060
02/05/2022 18:12:19
a is just any 'object', it doesn't really matter...
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:12:32
The group operation tells you how to handle multiplication between elements (making it work like 'normal multiplication')
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Moon Child
02/05/2022 18:12:52
Oh I see
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Shuri2060
02/05/2022 18:13:00
What's different is a group might not be abelian
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:13:09
But all cyclic groups are abelian
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Shuri2060
02/05/2022 18:13:14
Since you've just got powers of the same element
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Shuri2060
02/05/2022 18:13:41
So <a> represents the cyclic group generated by a
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help-10
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Shuri2060
02/05/2022 18:13:56
if a has order m (from the question) and b has order n
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:14:24
$$\langle a\rangle=\{e, a, a^2, \cdots, a^{m-1}\}$$ $$\langle b\rangle=\{e, b, b^2, \cdots, b^{n-1}\}$$
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Moon Child
02/05/2022 18:15:33
If a has an order m, does it also mean a^m=e?
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Shuri2060
02/05/2022 18:15:42
right.
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Shuri2060
02/05/2022 18:15:50
You can see that must be the case in these 2 groups.
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Shuri2060
02/05/2022 18:16:01
Note the group operation is 'borrowed' from G
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:16:11
If we're going by the question
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:16:22
We have the same operation, but just different domain
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ee650a1eb2184f5ca1221029aa080be4
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Moon Child
02/05/2022 18:16:47
I see
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TeXit
02/05/2022 18:17:34
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:17:37
looks better
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Shuri2060
02/05/2022 18:17:52
Alright, so now you want to think what the direct product of those 2 look like
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Shuri2060
02/05/2022 18:17:56
You are familiar with the definition of direct product?
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Moon Child
02/05/2022 18:18:09
Not really
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:18:22
you can think of it as like cartesian product
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:18:24
$\bR^2 = \{(a, b) : a, b \in\bR\}$
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517
help-10
ee650a1eb2184f5ca1221029aa080be4
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TeXit
02/05/2022 18:18:53
**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 18:18:57
Similar idea, but for groups
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1
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:19:10
$$A\times B = \{(a, b):a\in A, b\in B\}$$
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help-10
ee650a1eb2184f5ca1221029aa080be4
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TeXit
02/05/2022 18:19:12
**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 18:19:53
But in this case, we're doing this with order m and n?
1
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Shuri2060
02/05/2022 18:20:07
yh... but first of all you need to be familiar with how the group operation works
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Shuri2060
02/05/2022 18:20:17
it turns out exactly how you expect in some ways
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:20:39
$(a, b)(c, d) = (ac, bd)$
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help-10
ee650a1eb2184f5ca1221029aa080be4
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TeXit
02/05/2022 18:20:41
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:20:59
Note there are 3 different group operations going on here (but all 3 are referred to as 'multiplication)
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:21:43
$(a, b)\left(*_{A\cross B}\right)(c, d) = (a\left(*_{A}\right)c, b\left(*_{B}\right)d)$
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:21:54
hmm that's ugly 😂
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:22:31
If that makes sense?
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TeXit
02/05/2022 18:23:02
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:24:00
**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 18:24:10
These are 3 different binary operations in general.
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ee650a1eb2184f5ca1221029aa080be4
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Moon Child
02/05/2022 18:24:27
I'm a bit confused, which are from which?
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:24:38
I am using usual function notation to show
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1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:24:59
Or I could use different symbols if that makes it clearer
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Moon Child
02/05/2022 18:25:50
Yeah
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Shuri2060
02/05/2022 18:26:06
$*_1:A\to A$ \\ $*_2:B\to B$ \\ $*_3:A\times B\to A \times B$
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:26:25
Just treat them as any symbols
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TeXit
02/05/2022 18:26:49
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 18:27:06
$(a, b)*_3(c, d) = (a*_1c, b*_2d)$
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TeXit
02/05/2022 18:27:08
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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TeXit
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Shuri2060
02/05/2022 18:27:29
This is an important thing to note
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Shuri2060
02/05/2022 18:27:48
So $a *_1 c$ makes sense because a and c come from A
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:27:50
**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 18:27:52
And these are the operations of the three groups?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:27:56
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:27:57
Yes
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:28:06
Because of how we defined A x B
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:28:10
a, c come from A
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:28:14
b, d come from B
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:28:21
And by closure, ac is in A
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:28:24
bd is in B
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:28:32
so (ac, bd) is still in A x B
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:30:53
Ah yes
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ee650a1eb2184f5ca1221029aa080be4
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Moon Child
02/05/2022 18:31:08
I understand
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:31:37
So I'm going to use a small example
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:31:37
$$C_2\times C_3 = \{(a, b):a\in C_2, b\in C_3\}$$
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:32:08
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:32:28
$$\{(e, e), (a, e), (e, b), (a, b), (e, b^2), (a, b^2)\}$$
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 18:32:30
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 18:32:36
Can you see how this is the entire group?
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 18:33:34
How come there's no a^2?
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