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517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 510,789,298,321,096,700 | TeXit | 02/05/2022 18:56:40 | **Shuri2060** | 321445119171756033.png | 0 | 1 | null | null | null | null | null | ||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:56:42 | ok? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:56:49 | This is only possible since G is abelian | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:59:50 | Yes | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:00:32 | So now, what is the order of ab? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:00:39 | Remember gcd(a, b) = 1 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 19:03:50 | It's mn | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:03:58 | yes. | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:04:05 | What does this tell you about G? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 19:04:53 | The elements in G commute? | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:05:01 | You already knew that. G is abelian. | 0 | 1 | null | null | null | null | null | |||||
517 | 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. | 0 | 1 | null | null | null | null | null | |||||
517 | 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** | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 19:08:41 | Oh | 1 | 0 | null | null | null | null | null | |||||
517 | 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 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:09:00 | G = <ab> | 0 | 1 | null | null | null | null | null | |||||
517 | 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 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:09:08 | is ok | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:09:25 | unknown.png | 0 | 1 | null | null | null | null | null | |||||
517 | 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> | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:10:00 | make sense? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 19:11:12 | yes | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:11:42 | 321445119171756033.png | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:11:46 | Let's go with this example | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:11:52 | when order of a = 2 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:11:54 | order of b = 3 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:11:58 | So mn = 6 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:12:08 | So order of ab = 6 in G | 0 | 1 | null | null | null | null | null | |||||
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. | 0 | 1 | null | null | null | null | null | |||||
517 | 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? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:12:33 | Just think about mapping 1 element for now | 0 | 1 | null | null | null | null | null | |||||
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 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:13:22 | We want to **create** an isomorphism | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:13:28 | between G and <a> x <b> | 0 | 1 | null | null | null | null | null | |||||
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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:13:57 | Try thinking whats a good idea to try | 0 | 1 | null | null | null | null | null | |||||
517 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | 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 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:15:18 | A function | 0 | 1 | null | null | null | null | null | |||||
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? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:15:58 | unknown.png | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:16:00 | Because you have something like this | 0 | 1 | null | null | null | null | null | |||||
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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:16:15 | 1 is 'mapped' to 5 in my picture | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:16:27 | 321445119171756033.png | 0 | 1 | null | null | null | null | null | |||||
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 | 0 | 1 | null | null | null | null | null | |||||
517 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:17:07 | Just 1 element at a time | 0 | 1 | null | null | null | null | null | |||||
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> | 0 | 1 | null | null | null | null | null | |||||
517 | 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) | 0 | 1 | null | null | null | null | null | |||||
517 | 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 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:18:06 | No, from group G | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:18:13 | to this group <a> x <b> | 0 | 1 | null | null | null | null | null | |||||
517 | 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? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:18:37 | any | 0 | 1 | null | null | null | null | 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 | null | null | null | null | null | |||||
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 | 321445119171756033.png | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:19:13 | in here | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 19:19:26 | (a,e)? | 1 | 0 | null | null | null | null | 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 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:19:36 | f(a) = (a, e) | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:19:39 | Another element from G? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 19:19:49 | b | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:19:52 | f(b) = ? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:19:56 | Just take a guess | 0 | 1 | null | null | null | null | null | |||||
517 | 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 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 19:20:00 | (e,b) | 1 | 0 | null | null | null | null | 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 | null | null | null | null | null | |||||
517 | 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 | null | null | null | null | null | |||||
517 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:20:27 | You chose where a and b map to | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:20:30 | everything else must follow | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:20:42 | if we want it to be an isomorphism | 0 | 1 | null | null | null | null | null | |||||
517 | 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 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:20:59 | Good so far? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 19:21:03 | Mhm | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:21:06 | Then you can also check | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:21:07 | a^2 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:21:08 | b^2 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:21:14 | ab^2, etc. | 0 | 1 | null | null | null | null | null | |||||
517 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:21:37 | So we need to prove this idea works in general | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:21:51 | 321445119171756033.png | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:22:25 | $$\langle a\rangle \times \langle b\rangle= \{(a, b):a\in \langle a\rangle, b\in \langle b\rangle\}$$ | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 510,789,298,321,096,700 | TeXit | 02/05/2022 19:22:27 | **Shuri2060** | 321445119171756033.png | 0 | 1 | null | null | null | null | null | ||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:22:56 | $$ord(a) = m$$
$$ord(b) = n$$ | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 510,789,298,321,096,700 | TeXit | 02/05/2022 19:23:17 | **Shuri2060** | 321445119171756033.png | 0 | 1 | null | null | null | null | null | ||||
517 | help-10 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:24:50 | $ab$ generates G because its order is $mn$ | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 510,789,298,321,096,700 | TeXit | 02/05/2022 19:24:52 | **Shuri2060** | 321445119171756033.png | 0 | 1 | null | null | null | null | null | ||||
517 | help-10 | 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>? | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:25:55 | yes exactly | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | 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? | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 19:27:06 | I think you used them already | 0 | 1 | null | null | null | null | null |
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