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You can get the average of an average, but you would be averaging one number, so you would get the same result. You can get the average of a set of averages, though to make sense they will often need to be a weighted average of averages.
It is also an average. It is usually a better measure of the average value of the characteristic that is being measured.
No, a sum of averages is NOT as accurate as the average of the whole. For example: A=avg (1,10) = 5.5 B=avg (1, 1, 1, 1, 1) = 1 avg(A,B) = 3.25 [Average of averages] avg(1,1,1,1,1,1,10) = 2.29 [The original data set]
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
The properties of a subgroup would include the identity of the subgroup being the identity of the group and the inverse of an element of the subgroup would be the same in the group. The intersection of two subgroups would be a separate group in the system.