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The question's been studied in the category of groups, too. R. Hirshon proved in [1] that finite groups can always be cancelled in direct products.

Hirshon mentions some other sufficient conditions for the cancellation theorem to hold. For instance, he says that in the treatise by L. Fuchs there is a proof of the fact that infinite cyclic abelian groups can also be cancelled (provided that either $B$ or $C$ is a commutative group).

References

[1] R. Hirshon, On Cancellation in Groups, Amer. Math. Monthly. 76 (9) (1969), pp. 1037-1039.

2 added 57 characters in body

The question's been studied in the category of groups, too. R. Hirshon proved in 1 that finite groups can always be cancelled in direct products.

Hirshon mentions some other sufficient conditions for the cancellation theorem to hold. For instance, he says that in the treatise by L. Fuchs there is a proof of the fact that infinite cyclic abelian groups can also be cancelled (provided that either $B$ or $C$ is a commutative group).

References

[1] R. Hirshon, On Cancellation in Groups, Amer. Math. Monthly. 76 (9) (1969), pp. 1037-1039.

1

The question's been studied in the category of groups, too. R. Hirshon proved in 1 that finite groups can always be cancelled in direct products.

Hirshon mentions some other sufficient conditions for the cancellation theorem to hold. For instance, he says that in the treatise by L. Fuchs there is a proof of the fact that infinite cyclic abelian groups can also be cancelled.

References

[1] R. Hirshon, On Cancellation in Groups, Amer. Math. Monthly. 76 (9) (1969), pp. 1037-1039.