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The following question has been bothering me for sometime and any help will be greatly appreciated. Let me first fix notation. Let $G$ a group and for $a\in G$, let $\phi_a(x) = axa^{-1}, \forall x \in G$. You may take for granted that $\phi: G \to Aut(G)$, the map sending $a \to \phi_a$ is a group homomorphism.

Let $G$ be a normal subgroup of $M$. Denote by $C_M(G)$ the centraliser of $G$ in $M$. You may assume that that the function $f: G \times C_M(G) \to M$ defined by $f(g,m) = gm$ is a homomorphism of groups.

I would like to show the following:

(1) If $\phi$ is surjective then so is $f$

(2) If $\phi$ is injective, then so is $f$.

Thanks in advance!

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I think something's wrong here: of course $\,\phi_a\,$ is biejective: it is an inner automorphism! I think that in (1)-(2) you meant some other map, perhaps the one you denoted $\,a\to\phi_a\,$...or perhaps I missed something. – DonAntonio Dec 3 '12 at 3:10
    
Oh, I see nw: what is $\,\phi\,$ , anyway? – DonAntonio Dec 3 '12 at 3:11
    
You are absolutely right! $\phi$ is the map $a \to \phi_a$ – user44069 Dec 3 '12 at 3:17
    
from where to where? If it is from $\,G\, $ to $\,Inn(G)\,$ then it automatically is onto, but it if is to $\,Aut(G)\,$ not necessarily – DonAntonio Dec 3 '12 at 3:20
    
Corrected! Sorry once again! – user44069 Dec 3 '12 at 3:23

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