what are additive subgroups? i.e. 8Z + 12Z, what additive subgroup is it?
It looks like you want a very basic answer, so I'll give you a long answer with all the basic details, which you may not need. I figure it's always better to include the details than to possibly leave someone still confused. Z the integers a group under addition, since the following are all true: It has an identity element 0, where n+0 n for all integers n. There are inverses for every element: -n is the inverse of n (-n) + n 0 Addition is associative: (a+b) + c a + (b+c) 8Z would appear to be the integers which are a multiple of 8: . -16 -8 0 8 16 ,,, etc. All the above rules still apply. It still has the additive identity, 0, and an inverse for every number, and it's closed under addition (sum of any two elements is still in the subset). So it's a group in its own right, and since it's a subset of Z, it's a subgroup. 12Z would appear to be the integers which are a multiple of 12. 8Z + 12Z doesn't appear to be a standard notation to me. What do you mean by 8Z + 12Z? The union of the multiples of 8 and the multiples of 12 is not a subgroup (12 - 8 4 is not in it, for example). I think it's likely to refer to the subgroup generated by the elements in the union of these two subsets. That would be the multiples of the Greatest Common Divisor of 8 and 12, which is 4. So I think 8Z + 12Z refers to 4Z, the multiples of 4, which is a subgroup as described above. (If it were 8Z x 12Z, I would take that to be the cross product of the two subgroups. This would be ordered pairs of the form (8n, 12m), where n and m are integers.)
ur answers were very supportive. wat about z/6z is tis one is same as you described above. and i want the same level of explanation for finding number of group homomorphism from s3 to z/6z.
