# Residually finite and finitely many homomorphisms to any finite group implies Hopfian

From Groupprops

## Statement

Suppose is a residually finite group and is also a group having finitely many homomorphisms to any finite group. Then, is a Hopfian group.

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## Proof

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**Given**: A group that is resudally finite and has finitely many homomorphisms to any fixed finite group. A surjective endomorphism of .

**To prove**: is an automorphism.

**Proof**: We prove this by contradiction.

**ASSUMPTION THAT WILL LEAD TO CONTRADICTION**: We assume that is not an automorphism. Since it is surjective, it must fail to be injective, so that it must have a nontrivial kernel. Let be a non-identity element in the kernel of .

Step no. | Assertion/construction | Given data/assumptions used | Previous steps used | Explanation |
---|---|---|---|---|

1 | There is a surjective homomorphism for some finite group such that is not the identity element | is residually finite, is a non-identity element | [SHOW MORE] | |

2 | All the homomorphisms , with varying over the natural numbers, are pairwise distinct homomorphisms from to |
is surjective, is in the kernel of | Step (1) | [SHOW MORE] |

3 | There are only finitely many homomorphisms from to | has only finitely many homomorphisms to any finite group | Step (1) (specifically, that is finite) | Given direct. |

4 | We have the required contradiction. | Steps (2), (3) | [SHOW MORE] |