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Abstract Algebra

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When is Addition of Maps between Groups Commutative?

ABSALG-LSN7E2

Let $(G_1,+_1)$ and $(G_2,+_2)$ be groups. Define the set $M$ to be all functions from $G_1$ to $G_2$. Equip $M$ with the binary operation $\star$ where $f_1\star f_2$ is given by $g\mapsto f_1(g)+_2 f_2(g)$. It can be shown $M$ is a group. What additional condition is necessary for $M$ to be abelian?

A

$G_1$ is abelian

B

$G_2$ is abelian

C

$G_1$ and $G_2$ are both abelian

D

$M$ is always nonabelian

E

None of the above.