## Putnam Problem

**1997 Putnam-A4**

**Problem:** Let be a group with identity and a mapping such that

whenever

Prove there exists such that is a homomorphism.

**Proof:** It’s clear that if does exist that

So, we claim that is a homomorphism. To do this we first note that

and so by assumption

Thus,

(the center of the image). Thus, we then note that

where the parentheses are merely for indicative grouping, so then

but, using the fact that we see that

Namely,

But, by definition and from where the conclusion follows.

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