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The subgroup of (Z8, +) generated by [2] is... ?

We are asked to find the subgroup of the group of integers modulo 8 under addition generated by the element 2:


The elements of (Z8,+) are G={0,1,2,3,4,5,6,7} with 0 the identity element for the operation +.


We can generate all of the subgroups using addition modulo 8:


[0]={0}


[1]={0,1,2,3,4,5,6,7}=G


[2]={0,2,4,6}


[3]={3,6,1,4,7,2,5,0}=G


[4]={0,4}


[5]={5,2,7,4,1,6,3,0}=G


[6]={6,4,2,0}=[2]


[7]={7,6,5,4,3,2,1,0}=G


The subgroup generated by [2] is {0,2,4,6}


Note that this is a subgroup: there is an identity {0}, it has the associative property as integer addition is associative, it has the closure property, and every element has an inverse. (0 is its own inverse, 2+6=6+2=0, and 4 is its own inverse.)


Also note that the order ("size") of the subgroups are factors of 8, namely 1,2,4, and 8. You can also look at the gcf between the generating element and 8 and compare to the "size" of the generated subgroup.

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