Kernel of $\Bbb Z/2^k\Bbb Z\to \Bbb Z/2\Bbb Z$

Question

Just to be sure I'm not an idiot, the kernel of $$\Bbb Z/2^k\Bbb Z\to \Bbb Z/2\Bbb Z\\ 1\mapsto 1$$ is $\Bbb Z/2^{k-1}\Bbb Z$ right?

Answer 1

Almost. If I am being pedantic, then no. However, the kernel is isomorphic to $\Bbb Z/2^{k-1}\Bbb Z$. The kernel should be written as $\langle 2\rangle$ to emphasize that it's a subgroup of the domain, and not a group existing on its own right without being part of something bigger.

(At least until you get to category theory, where kernels truly are their own, separate objects, together with a homomorphism which emulates the "inclusion" you're used to. But then they are also defined not uniquely, but "up to unique isomorphism".)