Abstract
Abstract It is well known that the number of homomorphisms from a group to a group is divisible by the greatest common divisor of the order of and the exponent of . We study the question of what can be said about the number of homomorphisms satisfying certain natural conditions like injectivity or surjectivity. A simple non-trivial consequence of our results is the fact that in any finite group the number of generating pairs such that is divisible by the greatest common divisor of fifteen and the order of the group .
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