How to Determine the Order of an Element in a Factor Group?

[factor]

I'm currently studying factor groups in abstract algebra and needed some help understanding how to determine the order of an element in a factor group

Suppose I have Z (mod 12) / <4>. And I choose some random element from Z (mod 12) such as 5 or 7. How would I go about determining the order of 5 + <4> or 7 + <4> in the factor group. The way I understand is that in Z (mod 12) / <4> we collapse everything in <4> to the identity in Z (mod 12) so that 5 for instance would have order |{5,10,3,8}| = 4 because 8 is an identity in the factor group. Any help or correction on my admittedly poor understanding would be great.

 

[sutupidmath]

From a theorem, i believe, if the index of a normal group, N, in a group G is say, like in your case 4. then the order of the factro group G/N is 4. So, by lagranges theorem for every el in G/N, say C, C^4=N (identity). Now in your case, the order of Z_12/<4> is 4, so there will be 4 distinct cosets of <4> in Z_12, all of length 3, which makes me think, that the order of any of the el. in Z_12/<4> is going to be 3.

 

[jeane]

The way I understand it is list the elements of the factor group. <4>={0,4,8} then list the factors of the element you are trying to find the order of. 5, 10, 3, 8 Stop it took 4 times to get to an element in the factor group. Therefore the element has an order of 4.