Please Subscribe here, thank you!!! goo.gl/JQ8Nys Direct Products of Finite Cyclic Groups Video 2. How to determine if the direct product of finite cyclic groups is cyclic. Better examples than the first video.
For the example 4,7,17,25, the quick way of doing it in your head is to note that 7 and 17 are prime. 4=2*2 and 25=5*5, and 7,17,and 25 don't contain a factor of 2, and 4,7,and 17 don't contain a factor of 5.
C4 x C5 x C10 is not cyclic but is it of order 200? Oh no! Because is not Isomorphic to any cyclic group and therefore is not isomorphic to C200. What what is the order of these three cyclic groups multiplied as a whole?
Correct, it is not cyclic but it is of order 200. It's not cyclic because 4, 5, and 10 are not pairwise relatively prime since gcd(4, 10) = 2 != 1. It is of order 200 because |C_4 x C_5 x C_10| = |C_4|*|C_5|*|C_10| = 4*5*10 = 200. Note in general for finite groups G_1, G_2, ..., G_n we have |G_1 x G_2 x ... x G_n| = |G_1|*|G_2|*...*|G_n|.
+The Math Sorcerer What changes when you consider things such as C_6 x C_6? I have to list all possible orders of the group C_6 x C_6. Where the possible orders are the orders of elements of the group; I understand how to do this using Lagrange's Theorem. However, the orders of elements of C_6 can only be 1, 2, 3 or 6 and the solutions say that this is the same for C_6 x C_6. How come the order of C_6 x C_6 isn't 36, like with |C_4 x C_5 x C_10| = |C_4|*|C_5|*|C_10| = 4*5*10 = 200? Thanks for any help!
Sorry to bother you again but I have an other question, I hope you could explain me! For example: Z22 x Z21 x Z21 = (Z2 x Z11) x (Z7 x Z3) x ( Z7 x Z3) = Z2 x Z11 x Z7 x Z7 x Z3 x Z3. Now, could I conclude as Z22 x Z21 x Z21= Z2 x Z11 x Z49 x Z9 ?? Since 49 and 9 are prime powers, would this be a possible solution? Many thanks, Nataly x
And for example: the order of elements of Z24 are 1, 2 3 4 6 8 12 and 24. And so does Z3 x Z8 have the same order of elements, yes? But, Z2 x Z12 does only have the following order of elements: 1 2 3 4 6 8 and 12? 24 does not count because it is not part of Z12? Mmm... I hope you understand what I am try to ask :/
In general the order of an element (g_1, ..., g_n) in G_1 x ... G_n(assuming each g_i has finite order) is o((g_1, ..., g_n)) = LCM(o(g_1), ..., o(g_n)). So for example, in Z_12 we have o(8) = 3 and in Z_18 we have o(15) = 6, so in Z_12 x Z_18, the order of (8, 15) is LCM(3, 6) = 6
