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okeen
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Homework Statement
Let (G, °) be a group such that the mapping f from G into G defined by f(a) = a^(-1) is a homomorphism. Show that (G, °) is abelian.
The Attempt at a Solution
f(a) = a^(-1)
f(a^(-1)) = f(a)^(-1) = (a^-1)^-1 = a
in order for a group to be abelian it needs to meet the requirement a(i)*a(j) = a(j) * a(i)
° 1 a a^-1
1 1 a a^-1
a a a^2 1
a^-1 a^-1 1 a
since each side of the diagonal are the same then (G, °) is abelian.