# 1)      Consider the group  Zsub16 under addition. List all the elements of the subgroup <[6]> , and state its order.

mlehuzzah | Certified Educator

The identity element for `ZZ_n` (under addition) is [0]

A subgroup generated by one element is just the set of all multiples (or powers, if multiplicative notation) of that element.

The order of a group (or subgroup) is the number of elements in that group.  The order of an element is the smallest multiple (or power) of that element to get back to the identity element.

It turns out that for a group generated by just one element, the order of that element is the same as the order of the group.

[6] `!= [0]`

[6]+[6] = [12] `!= [0]`

[6]+[6] +[6]= [18] = [2] `!= [0]`

[6]+[6] +[6]+[6]= [24] = [8] `!= [0]`

[6]+ [6]+[6] +[6]+[6]= [30] = [14] `!= [0]`

[6]+ [6]+ [6]+[6] +[6]+[6]= [36] = [20]=[4]  `!= [0]`

[6]+[6]+ [6]+ [6]+[6] +[6]+[6]= [42] = [26]=[10] `!=` [0]

[6]+[6]+[6]+ [6]+ [6]+[6] +[6]+[6]= [48] = [32]=[16]=[0]

So the order of [6] is 8

Moreover, we can use this list to get an explicit description of <[6]>` `

<[6]> = { [6] , [12] , [2] , [8] , [14] , [4] , [10] , [0] }

Order doesn't matter, so a simpler way to write this set is:

<[6]> = { [0] , [2] , [4] , [6] ,[8] , [10] , [12] , [14]  }