What is the reason in each of these subgroups of order $2^n$, each element's lower $n$ bits are distinct & form all possible $4$-bit patterns?

Question

Let's take the example of the additive group $\Bbb Z/176\Bbb Z$

This has a subgroup of order $2^3$: $[0, 22, 44, 66, 88, 110, 132, 154]$

If you check the lower $3$ bits of each element in this subgroup, you will find $2^3$ distinct bit patterns covering each bit pattern from $0$ to $2^3$

Likewise this is the subgroup of order $2^4 = [0, 11, 22, 33, ...., 165]$

If you check the lower $4$ bits of each element in this subgroup, you will find $2^4$ distinct bit patterns covering each bit pattern from $0$ to $2^4$

Why is this so? Is there is name for this? Or is there a proof for how this is always true? And are these some conditions under which this is true?

Answer 1

It's because your examples consist of some (in the first case) or all (in the second case) multiples of $11$, and $11$ is relatively prime to $16$. So in the second case, you get all possible remainders modulo $16$. In the first case the last bit is always $0$ but the previous 3 bits give all possibilities modulo $8$.

Something like this will work in any similar case. For example if you have the first $9$ multiples of $11$ and you write them in base $3$, you will find that the last two digits give all possible combinations.