Let $H=\{1,h\}$ and $A=\{0,a\}$ be groups, and $\pi:H\rightarrow \text{Aut}(A)$ be the trivial homomorphism. I have found $FS(H,A,\pi)=\{f_0,f_1\}$ and $IFS(H,A,\pi)=\{f_0\}$

where $f_0(1,1)=f_0(1,h)=f_1(h,1)=f_0(h,h)=0$ and

$f_1(1,1)=f_1(1,h)=f_1(h,1)=0,f_1(h,h)=a$

So I know that we only get two extensions of $H$ by $A$ which is obvious enough, but how can I actually describe the group isomorphic to $Z_4$ by $\frac{FS(H,A,\pi)}{IFS(H,A,\pi)}=EXT(H,A,\pi)=\{f_0,f_1\}$?

Is there a way to find the group presentation for the group by knowing $EXT(H,A,\pi)$?

Question

Let $H=\{1,h\}$ and $A=\{0,a\}$ be groups, and $\pi:H\rightarrow \text{Aut}(A)$ be the trivial homomorphism. I have found $FS(H,A,\pi)=\{f_0,f_1\}$ and $IFS(H,A,\pi)=\{f_0\}$

where $f_0(1,1)=f_0(1,h)=f_1(h,1)=f_0(h,h)=0$ and

$f_1(1,1)=f_1(1,h)=f_1(h,1)=0,f_1(h,h)=a$

So I know that we only get two extensions of $H$ by $A$ which is obvious enough, but how can I actually describe the group isomorphic to $Z_4$ by $\frac{FS(H,A,\pi)}{IFS(H,A,\pi)}=EXT(H,A,\pi)=\{f_0,f_1\}$?

Is there a way to find the group presentation for the group by knowing $EXT(H,A,\pi)$?

zzr0ck3r · Answer

@joemath314159