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Abstract Algebra

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Moderate

Counting Automorphisms of $\mathbb{Z}/20\mathbb{Z}$

ABSALG-HVYKEI

Let $\mathbb{Z}/20\mathbb{Z}$ be the additive group of integers mod $20$. (The notation $\mathbb{Z}_{20}$ for this group is also common.)

1) The number of group homomorphisms from $\mathbb{Z}/20\mathbb{Z}$ to itself is

.

2) The number of surjective group homomorphisms from $\mathbb{Z}/20\mathbb{Z}$ to itself is

.

3) The number of injective group homomorphisms from $\mathbb{Z}/20\mathbb{Z}$ to itself is

.

4) The number of group automorphisms from $\mathbb{Z}/20\mathbb{Z}$ to itself is

.