Question: Let $\lambda$ and $\lambda\_L$ be the values of the function $f(x,y)$ at the optimum for problems 
\begin{align}
\lambda=\max\_{x}\min\_{y}f(x,y) 
\end{align}
\begin{align}
\lambda\_L=&\max\_{x}\min\_{y} f(x,y) \\& \mbox{s.t.}~~g(y)=c
\end{align}
respectively. We can assume that $x$ and $y$ comes from the same domains in both problems. Is $\lambda\geq \lambda\_L$?

Answer: 
You can embed the Klein 4-group (which is not cyclic, since it contains no element of order $4$!) into $S\_4$ in four ways: 

$$\langle (12),(34)\rangle=\{1,(12),(34),(12)(34)\} $$

$$\langle (13),(24)\rangle=\{1,(13),(24),(13)(24)\} $$
$$\langle (14),(23)\rangle=\{1,(14),(23),(14)(23)\} $$
$$\langle(12)(34),(14)(23)\rangle=\{1,(12)(34),(14)(23),(13)(24)\}$$

but only the last subgroup happens to be normal.
An easy way to do this is just to examine the permutations of order 2, since all elements of the Klein-4 group have order 2. These are just the set of all transpositions and disjoint transposition.

Note that the transpositions alone do not form a normal subgroup, so we have all of the pairs of disjoint transpositions: 
$$\{(),(12)(34),(13)(24),(14)(23)\}.$$