For groups G and H, let F(G,H) ={f:G→H;f is a function}. (a) Let G=R−{0} and H= \mathbb{Z}. Show that F(G,H) is a group
Posted: Tue Sep 07, 2021 7:40 am
For groups G and H, let F(G,H) ={f:G→H;f is a function}.
(a) Let G=R−{0} and H= \mathbb{Z}. Show that F(G,H) is a group
with the operation∗defined by (f∗g)(x) =f(x)+g(x)∀x∈G.Is this group
abelian?
(b) Now we will generalize the previous part. Let
(G, \textcircled{$\star$} ) and (H, \textcircled{$\dot$})
be groups. (The operations are arbitrary,not necessarily
multiplication.) Prove that F(G,H)is a group with the
operation∗defined by(f∗g)(x) =f(x) \textcircled{$\dot$}
g(x) ∀x∈G.
(c) Find necessary and sufficient conditions for F(G,H), as in
part (b), to be abelian.
(a) Let G=R−{0} and H= \mathbb{Z}. Show that F(G,H) is a group
with the operation∗defined by (f∗g)(x) =f(x)+g(x)∀x∈G.Is this group
abelian?
(b) Now we will generalize the previous part. Let
(G, \textcircled{$\star$} ) and (H, \textcircled{$\dot$})
be groups. (The operations are arbitrary,not necessarily
multiplication.) Prove that F(G,H)is a group with the
operation∗defined by(f∗g)(x) =f(x) \textcircled{$\dot$}
g(x) ∀x∈G.
(c) Find necessary and sufficient conditions for F(G,H), as in
part (b), to be abelian.