Exercise
Precisely three of the four assertions below regarding an arbitrary group G are equivalent to each other. Which is the outsider?
L = R.
C = R.
G is commutative.
C is trivial (meaning: Ker(C) = G).
No.
L = R means that, for all
g, x 
G,
we have gx = xg, which is equivalent to saying that G is commutative.
G,
we have gx = xg, which is equivalent to saying that G is commutative.
Yes.
C = R means that, for all
g, x G,
we have gxg-1 =
xg-1, which amounts to
g = e. Hence this condition says that G is the trivial group.
This is not equivalent to saying that G is commutative.
G,
we have gxg-1 =
xg-1, which amounts to
g = e. Hence this condition says that G is the trivial group.
This is not equivalent to saying that G is commutative.
No, see the first alternative.
No.
Ker(C) = G means
that, for all
g, x G,
we have gxg-1 = x,
which is equivalent to G being commutative.
G,
we have gxg-1 = x,
which is equivalent to G being commutative.