What does it mean for a function $g$ to be a permutation that fixes the image of $f$ pointwise?

Question

In van Lint's and Wilson's Combinatorics, the following question below asks:

Now what does it mean for $G$ to be a permutation that fixes the image of $f$ pointwise? Can anyone provide an example for some small values that elaborates what is going on?

Answer 1

$g:\{1, ..., n\}\rightarrow \{1, ..., n\}$ being a permutation only means that is a bijective function. On the other hand, $g$ fixing the image of $f$ pointwise, means that we have for all $x\in Im (f)\subset \{1, ..., n\},\: g(x)=x$.

Answer 2

To give an example, first denote $[n]:=\{1,2,\dots,n\}$ to be the set of the first $n$ positive integers.

Let $f:[5]\to[10]$ be a function such that $f(k) = 2k$. Note that the image of $f$ is $\{2,4,6,8,10\}$. Then $g:[10]\to[10]$ defined by $g = (1~3~5~7~9)$ is a permutation of $[10]$ that fixes the image of $f$ point-wise.

If we were to define $g$ differently, as, say, $g = (1~3~5~7~9)(2~4~6~8~10)$ then we would say $g$ fixes the image of $f$ set-wise but not point-wise.

Answer 3

There are two reasonable interpretations of the statement "the function $g : A \to A$ fixes the subset $S \subseteq A$." Namely:

1. $g(s) \in S$ for all $s \in S$, which can also be written as $g(S) \subseteq S$,

2. $g(s) = s$ for all $s \in S$

When most people write "$g$ fixes $S$" they usually mean the first interpretation. The second interpretation, therefore, requires a new name and that is "fixes pointwise." Instead of saying "$g$ fixes $S$ pointwise," one could equivalently say that $g$ restricts to the identity on $S$.

For example, let $A = \{1,2,3,4\}$ and $S = \{1,2\}$ then the permutation $h$ which swaps $1$ and $2$ "fixes" $S$ because $h(1) \in S$ and $h(2) \in S$ and so $h(S) \subseteq S$. On the other hand, $h$ does not fix $S$ pointwise because the point $1$ is not fixed, rather $h(1) = 2$.