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517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:33:40 | C2 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:33:41 | is just | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:33:43 | {e, a} | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:34:49 | All good? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:35:01 | Ah yes | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:35:09 | So let's stick with this example | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:35:15 | So in the original question let's say m = 6 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:35:31 | So I now have <a> x < b> | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:35:36 | 321445119171756033.png | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:35:50 | a = 2, b = 3 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:35:54 | and they are relatively prime. | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:36:12 | You now need to show this group must be isomorphic to the original group | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | 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 | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:36:33 | ok | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:36:39 | 321445119171756033.png | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | 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? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:37:27 | Well I understand {e,a} in C2 and {e,a,b} in C3? | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:37:31 | $$C_2 = \{e, a\}$$
$$C_3 = \{e, b, b^2\}$$ | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:37:39 | Or is it e b, and b2 | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 510,789,298,321,096,700 | TeXit | 02/05/2022 18:37:44 | **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 18:37:47 | Oh | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:37:49 | C_n is the cyclic group of order n | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:37:53 | There is only 1 generator | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:37:55 | for cyclic groups | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:38:17 | You are familiar with that right? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:38:59 | Then everything must be generated by it. | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:39:10 | Oh okay | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:39:45 | 321445119171756033.png | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:39:48 | 321445119171756033.png | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:40:13 | Mhm | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:40:26 | So to familiarize yourself with this | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:40:33 | What is (a, b)^2 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:41:15 | Written in this way? Or? | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:41:25 | You can go step by step | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:41:28 | So first | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:41:36 | (a, b)^2 = (a, b)(a, b) | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:41:59 | (a+a,b+b)? | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:42:14 | That all depends on the group operation | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:42:20 | but we usually prefer to write like multiplication | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:42:29 | $(a, b)^2 = (a, b)(a, b) = (a^2, b^2)$ | 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 18:42:31 | So like this. | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 510,789,298,321,096,700 | TeXit | 02/05/2022 18:42:38 | **Shuri2060** | 321445119171756033.png | 0 | 1 | null | null | null | null | null | ||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:43:05 | For the general case it will be written like this (the multiplication sign is omitted, but they **are** different) | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:43:19 | Oh okay | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:43:27 | 321445119171756033.png | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:43:33 | But then, this must correspond to one of these | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:43:34 | which? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:44:01 | (e,b^2) | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:44:06 | right | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:44:10 | because a^2 = e | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:44:15 | Mhn | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:44:26 | For clarity, sometimes it might be written like this | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:44:35 | $$\{(e_A, e_B), (a, e_B), (e_A, b), (a, b), (e_A, b^2), (a, b^2)\}$$ | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 510,789,298,321,096,700 | TeXit | 02/05/2022 18:44:38 | **Shuri2060** | 321445119171756033.png | 0 | 1 | null | null | null | null | null | ||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:44:45 | Because e_A and e_B are not necessarily the same | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:44:52 | They are just the identity elements from A and B | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:45:07 | 321445119171756033.png | 0 | 1 | null | null | null | null | null | |||||
517 | 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. | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | 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 ? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:46:26 | (e,e)? | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:46:34 | yep | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:46:40 | I think you get how this group works | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:46:48 | I do now | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:46:50 | unknown.png | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:46:54 | So the question... | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:47:08 | You need to show this thing is isomorphic to G | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:47:34 | Do you remember how an isomorphism works, roughly? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:47:38 | Just the main concept | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:48:09 | Show it's bijective and morphic | 1 | 0 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:48:16 | yah but the main concept is | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:48:19 | If we have 2 groups | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:48:21 | X and Y | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:48:24 | and they are isomorphic | 0 | 1 | null | null | null | null | null | |||||
517 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:48:47 | And the group will still behave exactly the same | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:48:57 | Just different labels (and group operation) | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:49:17 | f(xy)=f(x)f(y) | 1 | 0 | null | null | null | null | null | |||||
517 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:50:26 | You could focus on group G first. | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:50:37 | It tells you $\gcd(m, n) = 1$ | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 510,789,298,321,096,700 | TeXit | 02/05/2022 18:50:39 | **Shuri2060** | 321445119171756033.png | 0 | 1 | null | null | null | null | null | ||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:50:50 | a has order m
b has order n | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:51:23 | Hmm | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:52:00 | Thinking 😂 | 0 | 1 | null | null | null | null | null | |||||
517 | 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? | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 589,206,850,839,380,000 | Moon Child | 02/05/2022 18:54:31 | I'm no sure | 1 | 0 | null | null | null | null | null | |||||
517 | 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 | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:55:05 | But think about the element $ab$ | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 510,789,298,321,096,700 | TeXit | 02/05/2022 18:55:07 | **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 18:55:56 | If this helps, these are the problems that the question hints about | IMG_20220205_155537.jpg | 1 | 0 | null | null | null | null | null | ||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:56:06 | Right exactly | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:56:07 | $(ab)^2 = (ab)(ab) = a^2b^2$ | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 510,789,298,321,096,700 | TeXit | 02/05/2022 18:56:09 | **Shuri2060** | 321445119171756033.png | 0 | 1 | null | null | null | null | null | ||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:56:15 | First of all, the group is abelian | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:56:19 | So when we take powers of ab | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:56:23 | We can rearrange like this | 0 | 1 | null | null | null | null | null | |||||
517 | help-10 | ee650a1eb2184f5ca1221029aa080be4 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 18:56:37 | $$(ab)^k = a^kb^k$$ | 0 | 1 | null | null | null | null | null |
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