Is $f\colon\mathbb Z_{26}\to\mathbb Z_{26}$ a permutation? $f(a)=11a \pmod{26}$

Question

Is $f\colon\mathbb Z_{26}\to\mathbb Z_{26}$ a permutation? $$f(a)=11a\pmod{26}$$ Note: it must be one-to-one and onto.

I'm really struggling with how to start this question. Any help would be greatly appreciated!

score 0 · Accepted Answer · answered Apr 09 '17 at 03:36

Hint 1: If $X$ is a finite set, a function $f : X\to X$ is a bijection if and only if it is an injection. $\Bbb Z_{26}$ is certainly finite, so it suffices to check of $a\mapsto 11a$ is injective.

Hint 2: $a\mapsto 11a$ is a homomorphism $\Bbb Z_{26}\to\Bbb Z_{26}$, and a homomorphism is injective if and only if the kernel is $0$.

score 0 · Answer 2 · answered Apr 09 '17 at 03:37

0

Hint: $11\cdot 19 \equiv 1 \mod 26$. Can you use this to come up with an inverse function?

answered Apr 09 '17 at 03:37

Hayden

16,737

Is $f\colon\mathbb Z_{26}\to\mathbb Z_{26}$ a permutation? $f(a)=11a \pmod{26}$

2 Answers2