Exercise 4.5.8 (Sylow $2$-subgroups of $S_5$)

Question

Exhibit two distinct Sylow $2$-subgroups of $S_5$ and an element of $S_5$ that conjugates one into the other.

Richard Ganaye · Accepted Answer

\begin{proof} 
Consider the subgroup of $S_5$ given by
$$H = \langle (3\ 4), (1\ 2), (1\ 4)(2\ 3) angle.$$
Explicitly
$$H = \{(), (1\ 2), (1\ 4)(2\ 3), (1\ 3\ 2\ 4), (3\ 4), (1\ 2)(3\ 4), (1\ 4\ 2\ 3), (1\ 3)(2\ 4) \},$$
Then $|H| = 8$, so $H$ is a $2$-Sylow of $S_5$.
If $\sigma = (1\ 2\ 3 \ 5)$, then
$$K = \sigma H \sigma^{-1} =  \langle (2\ 3), (4\ 5), (2\ 4)(3\ 5) angle$$
is also a  $2$-Sylow of $S_5$, distinct of $H$, since $(2\ 4)(3\ 5) 
ot \in H$.
\end{proof}

With GAP:
\begin{verbatim}
gap> G := SymmetricGroup(5);;

gap> T := Subgroup(G,[]);;

gap> L := IntermediateSubgroups(G,T).subgroups;;

gap> for H in L do
>     if Order(H) = 8 then
>         Print(H, "
");
>     fi;
> od;

Group( [ (4,5), (2,3), (2,4)(3,5) ] )
Group( [ (3,5), (2,4), (2,3)(4,5) ] )
Group( [ (3,4), (2,5), (2,3)(4,5) ] )
Group( [ (1,4), (2,3), (1,3)(2,4) ] )
Group( [ (1,3), (2,4), (1,4)(2,3) ] )
Group( [ (3,4), (1,2), (1,4)(2,3) ] )
Group( [ (1,4), (3,5), (1,3)(4,5) ] )
Group( [ (1,3), (4,5), (1,4)(3,5) ] )
Group( [ (3,4), (1,5), (1,4)(3,5) ] )
Group( [ (1,4), (2,5), (1,5)(2,4) ] )
Group( [ (1,5), (2,4), (1,4)(2,5) ] )
Group( [ (4,5), (1,2), (1,4)(2,5) ] )
Group( [ (1,5), (2,3), (1,3)(2,5) ] )
Group( [ (1,3), (2,5), (1,5)(2,3) ] )
Group( [ (3,5), (1,2), (1,5)(2,3) ] )

gap> List(Group( [ (3,4), (1,2), (1,4)(2,3) ] ));

[ (), (1,2), (1,4)(2,3), (1,3,2,4), (3,4), (1,2)(3,4), (1,4,2,3), (1,3)(2,4) ]
\end{verbatim}

Exercise 4.5.8 (Sylow $2$-subgroups of $S_5$)

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Exercise 4.5.8 (Sylow $2$-subgroups of $S_5$)

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