.....of groups. I am trying to study group extension theory, and find there is something I don't quite get. I wonder if anyone can help.

Suppose

is a group and

. Then elementary group theory induces the quotient group

, with

as the identity, and a surjection

, No problems here, I believe.

The SES of homomorphisms

is now defined by my text as

which is OK too, since I am interpreting the "1" as a multiplicative unit (or possibly the identity matrix) so

and

are the obvious inclusion maps,

is the surjection above and

is the usual trivial homomorphism. Is this fair?