= Endomorphism of a finite cyclic group
{title2=$f_a(k)=ak\pmod n$}
= Endomorphisms of a finite cyclic group
{synonym}
Every <endomorphism> of the additive <finite cyclic group> $C_n=\mathbb Z/n\mathbb Z$, for a positive <integer> $n$, has the form $f_a(k)=ak\pmod n$, determined by $a=f_a(1)$. This is the self-map case of <homomorphism from a finite cyclic group>: repeated <addition> forces the value at every element. Conversely, each choice of $a$ respects <addition> and gives a well-defined map on <congruence classes>.
Let $d=\gcd(n,a)$, the <greatest common divisor>. The <image of a group homomorphism> is the <cyclic subgroup> generated by $a$, with size $n/d$ by <order of an element of a finite cyclic group>. The <kernel of a group homomorphism> is
$$
\ker f_a=\{0,n/d,2n/d,\ldots,(d-1)n/d\},\qquad |\ker f_a|=d.
$$
Indeed, $ak=0\pmod n$ is equivalent to $n/d$ dividing $k$, after cancelling $d$ and using <coprimality>. Consequently $f_a$ is a <group automorphism> exactly when $a$ and $n$ are <coprime>. In that case a <modular inverse> $b$ of $a$ gives $f_a^{-1}=f_b$. This connects reversible self-maps directly to the <generators of a finite cyclic group>.<Group Theory (v4.01) by James S. Milne>[Chapter 3, Automorphisms of cyclic groups]{ref}
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