Abstract Algebra. I'm a little bit unclear on the process of finding subgroups.

[BamBam215]

Let's say I have the group of integers, Z(16), under addition. I need to find all the elements of the subgroup <[6]>.

I don't really understand the process of finding each element?

 

[tikwanleap]

Originally posted by: BamBam215
Let's say I have the group of integers, Z(16), under addition. I need to find all the elements of the subgroup <[6]>.

I don't really understand the process of finding each element?

uhh... yeah really. Things that make you go Hmmm???

 

[Soccer55]

You have to find a set of elements that satisfy the requirements for being a group. So for example, in Z16, one subgroup would be {0,2,4,6,8,10,12,14}. If you don't see this right away, take each element of Z16 and see what group it generates. For example, <4> = {0,4,8,12} since 4+4=8, 8+4=12, 12+4=16=0 mod 16, 0+4=4. On the other hand <3>=Z16. You'll start to notice a common theme among the elements that generate a subgroup in Z16 and it will apply to Zn for any n (Hint: look at gcds). Hope this helps.

-Tom

EDIT: In my half-awake state, I buried in this explanation the way you would find the elements for <6> in Z16. You just use the operation of Z16 (addition) and you keep adding 6 to itself until you don't get anything new. Then all of the numbers you have produced by adding 6 to itself will make up the subgroup.

 

[Muzzan]

<a> is defined as {a^k; k \in Z}. It can be shown that if a has finite order (say n), then <a> = {a^0, ..., a^(n - 1)} (and it can also be shown that every element of a finite group has a finite order). Apply this to your problem (and don't forget to translate everything into additive notation).