# Order of Hx in Factor Group

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• Nov 29th 2007, 09:05 PM
temp31415
Order of Hx in Factor Group
Another Factor Group question

Question: Let G = U17 and H = <[4]> be the cyclice subgroup generated by [4] in G/H. Find the order of Hx where x = 2 or 3.

G = {1,2,3,...,16}
H = {4,8,12,16}
|G/H| = |G|/|H| = 16/4 = 4
o(x)|4 (the order of an element must divide the order of the group). so o(x) = 1,2, or 4
if x = 2 then o(H + [2]) != 1 and 2(H + [2]) = H + [4] = H + [0] so o(H + [2]) = 2
if x = 3 then o(H+ [3]) != 1 or 2 and 3(H + [3]) = H + [12] = H + [0] so o(H + [3]) = 4

Thanks in advance!
• Nov 29th 2007, 09:15 PM
ThePerfectHacker
Quote:

Originally Posted by temp31415
Another Factor Group question

Question: Let G = U17 and H = <[4]> be the cyclice subgroup generated by [4] in G/H. Find the order of Hx where x = 2 or 3.

You got it wrong.
$G=\mathbb{Z}_{17}^{\text{x}}=\{1,...,16\}, H=\{4,16,13\}$
• Nov 29th 2007, 09:32 PM
temp31415
I think that makes sense. 13 is in H because 64 mod 17 = 13?
and 1 and 8 aren't in H because they aren't powers of 4?

Thanks!
• Nov 29th 2007, 09:34 PM
ThePerfectHacker
Remember this is not addition, it is multiplication. Do not get confused.