What do $0$ and $1$ mean in a extension (exact sequence) of group?

Question

I'm self-studying homological algebra but stuck in a little place. What does $0$ and $1$ mean by the author? (I didn't read the book from start to end.) Does it mean the trivial group? Then why we write in different symbols?

Answer 1

My immediate suspicion: $K$ is abelian, written additively, and the trivial (sub)group is $0$. $Q$ is just a group, and is written multiplicatively, making the trivial (quotient) group $1$.

Answer 2

Both $0$ and $1$ are generaly used to represent the trivial group. Usually, $0$ is used in an abelian context (where the group operation is written additively and the identity element as $0$) and $1$ is commonly used in a non (necessarily) abelian context (the group operation is written multiplicatively, so the identity element is $1$).

Since $0$ can be identified with a subgroup of $K$ and $K$ is abelian, they write it as $0$. In general, you can use both but is not very common to see the two notations mixed.