# Continuous Maps, Homomorphisms and Cyclic Groups

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Let f: S^n --> S^n be a continuous map.

Consider the induced homomorphism f*: H~_n (S^n) --> H~_n (S^n), where

H~_n is a reduced homology group. Then from the fact that H~_n (S^n) is an infinite

cyclic group, it follows that there is a unique integer d such that

f*(u) = du for any u in H~_n (S^n).

How exactly does "there is a unique integer d such that..." follow from

"H~_n (S^n) is an infinite cyclic group"?

https://brainmass.com/math/group-theory/continuous-maps-homomorphisms-cyclic-groups-88731

#### Solution Preview

Call H~_n(S^n) G for short.

Take a generator g of the group G, so G = {1, g, g^2, g^3 ,...., g^{-1}, g^{-2},... ...

#### Solution Summary

Continuous Maps, Homomorphisms and Cyclic Groups are investigated.

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