How does the index of this subgroup is a power of 2?

Question

I am reading an article about coset codes (for answering this question having knowledge about these codes and lattice theory is not necessary) which are defined by $(\Lambda ,\Lambda ',C)$ in which $\Lambda$ is an $N$- dimensional Lattice Constellation (an $N$ dimensional lattice is a group under ordinary $N$ tuples addition), $\Lambda '$ is its sub-lattice and $C$ is an encoder. It has been said that in these codes we are generally interested in binary lattices, and it means that for some $n$ we have : $Z^{N}/\Lambda /\Lambda '/2^{n}Z^{N}$, in which $2^{n}Z^{N}$ is the lattice of all $N$ tuples of integer multiples of $2^{n}$. It has said that from the above relation we conclude that the index of $\Lambda '$ in $\Lambda$ is a power of $2$ which I cannot figure it out. Can anyone explain why it is a power of $2$ ?

Answer 1

$[Z^{N}:2^{n}Z^{N}]=[Z^{N}:\Lambda][\Lambda:\Lambda '][\Lambda ':2^{n}Z^{N}]$

and we know $[Z^{N}:2^{n}Z^{N}]=2^{nN}$, hence we conclude

$[\Lambda:\Lambda '] | 2^{nN}$

so it must be a power of $2$

Answer 2

Hint: index is transitive, i.e. $[G:K]=[G:H][H:K]$.