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This question has several questions within it pertaining to my title.
Consider the subset S={[0],[2],[3]} of the modulo 8 to be considered as the additive group modulo 8.
1) Determine the set S+S?
2)Is it a subset?
3)If H was a subgroup of modulo 8, what do you expect H+H to be?
4)Determine the subgroup generated by S, i.e., the smallest subgroup of modulo 8 contains S?
Here is my attempt to try to do it.Without doing the 1st question properly,I cant do the others.
My set modulo 8 is {[0],[1],[2],[3],[4],[5],[6],[7]}. Is using this set the correct way to determine S+S?

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    what is your definition of the operation $S+S$? – 2017-02-11
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    Its a subset. Hence, subset S=[0],[2],[3] – 2017-02-11
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    you are adding to subsets together by additive modulo 8 – 2017-02-11
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    Mathjab please! – 2017-02-11
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    me? sorry. i was rushing – 2017-02-11

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Assuming that $W=S+S$ means that for $W=\{ ((s+t)\bmod 8) \forall s,t\in S\}$ then $S+S=\{[0],[2], [3],[4],[5],[6]\}$.

We know it must be a subset (before even enumerating its members) because the additive group modulo $8$ is closed.

If $H$ is a subgroup, it must be closed: what does that say about $H+H$?

For the last question, consider that $3$ and $8$ are coprime.

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    based on your answer, H+H is a subset of itself. Am I right in assuming that? – 2017-02-11
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    Kind of correct... $H+H$ should be $H$. – 2017-02-11
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    Okay One more thing, how did you come up with {[0],[2],[3],[4],[5],[6]} as S+S? – 2017-02-11
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    I know you got to add [0],[2],[3] to something – 2017-02-11
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    $\{[0]\}+\{[0],[2],[3]\} = \{[0],[2],[3]\}$ ----- $\{[2]\}+\{[0],[2],[3]\} = \{[2],[4],[5]\}$ ----- $\{[3]\}+\{[0],[2],[3]\} = \{[3],[5],[6]\}$ - take the union – 2017-02-11
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    Oh. Okay. Thank you. For H+H, I got to use the subset [0],[2],[3] still? – 2017-02-11
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    No, you are given that $H$ is a subgroup. For example, $\{[0],[4]\} is a subgroup (not the only one) – 2017-02-11
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    so [0],[2],[4],[6] is the subgroup of modulo 8 for H but [3],[5] are not – 2017-02-11
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    There are $3$ proper subgroups of your group mod $8$; the other one is just the trivial one of $\{[0]\}$, but in each case $H+H=H$. So you don't know which subgroup you are dealing with but the answer is the same, due to group properties. – 2017-02-11