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For example,

For U(15) = {1,2,4,7,8,11,13,14} under multiplication modulo 15. This group has order 8. To find the order of the element 7, say, we compute the sequence 7^1= 7, 7^2= 4, 7^3= 13, 7^4= 1, so |7|=4. But, how do you find that 1,2,4,7,8,11,13,14 are part of U(15)? if one did not give you the set, how would you be able to find the order of U(15)? or even the order of 7 (|7|)?

A Bigger problem I have,

How do you find order for matrices?

like if A= [0, -1, 1, 0] what is the order of |A|?

Please help!!! I am so confused on how to find order! I reallly appreciate it!! Thanks!