This post is a brief application of a result discussed in the last post about the existence of odd ways to map the fundamental group of the Hawaiian earring onto an arbitrary finite group :
Theorem 1: Let be any non-trivial finite group and be a loop going once around the n-th circle of the Hawaiian earring in the clockwise direction. There are uncountably many surjective group homomorphisms mapping to the identity element for every .
Since the kernel of each of these homomorphisms contains the infinite free group generated by the classes , there is clearly a connection to the harmonic archipelago .
Let’s fix a non-trivial finite group and show that Theorem 1 also holds for the Harmonic archipelago.
Recall that the canonical inclusion induces a surjective homomorphism (see Corollary 2 of this post). Moreover, is the conjugate closure of the free group generated by the elements , .
Of course, we have . So for every surjective homomorphism satisfying , the inclusion holds and we get a unique surjective homomorphism such that .
Altogether, induces an injection
which is pre-composition by and we have when is one of the (uncountably many) surjective homomorphisms guaranteed to exist by Theorem 1. This is enough to serve as the proof of the main theorem of this post.
Theorem 2: Let be any non-trivial finite group and be a loop going once around the n-th circle of viewed as a subspace of . There are uncountably many surjective group homomorphisms mapping to the identity element for every .
Certainly then, we have uncountably many (overall) homomorphisms from to .
Corollary 3: For any non-trivial finite group , the set of group homomorphisms is uncountable.
Theorem 2 and Corollary 3 are in stark contrast to the fact that the only homomorphism to the additive group of integers is the trivial homomorphism (See Theorem 1 of The harmonic archipelago group is not free).