I don't really understand what you are trying to do, but here's what I think you mean: That in the ring of integers modulo 10, 2*8=1 mod(10), but this is not true since the inverse of 2 is 5, and 2*8=16=6 mod(10). Also, I don't know what H has to do here, since operations on H are inherited from the whole ring.

Now if you mean that 2+8=0 mod(10) then this is indeed true because 10 divides 10, and hence 2 is the inverse of 8 in the ADDITIVE subgroup H.