Lang's Algebra Ch 1 Problem 12(c): Ambiguous Group Law

[eastside00_99]

I am working the problems in Lang's Algebra. I am on number 12(c) on Chapter 1 [revised third edition], it states

let H and N be groups and let f: H --> Aut(N) be given homomorphism Define G = NxH with the law

(a,b)(a',b') = (a'f(b)a',bb').

the problem is that f(b) is a member of Aut(N) and so the definition is ambiguous at best. I mean I suppose it could mean f(b) applied to a' (or even applied a). But this doesn't fit the notation of the rest of the book. Any comments on this?

 

[morphism]

(Should that be "(a f(b) a', bb')" instead?)

I'm pretty sure it's supposed to mean f(b) applied to a'. It looks like the exercise is attempting to walk you through the construction of the semidirect product of two groups.

 

[eastside00_99]

(Should that be "(a f(b) a', bb')" instead?)

I'm pretty sure it's supposed to mean f(b) applied to a'. It looks like the exercise is attempting to walk you through the construction of the semidirect product of two groups.

yeah i meant a instead of a' up there. Ok, I guess that has to be what it is then. Thanks.