Do there exist 2-sylow subgroups of $S_4\times S_3$ that are normal?
Do there exist 3-sylow subgroups of $S_4\times S_3$ that are normal?
Thank you for helping!
Do there exist 2-sylow subgroups of $S_4\times S_3$ that are normal?
Do there exist 3-sylow subgroups of $S_4\times S_3$ that are normal?
Thank you for helping!
No. The list of normal subgroups can easily be calculated, for instance using GAP. Notice none of the normal subgroups has the correct order to be a Sylow subgroup.
gap> K := SymmetricGroup(3); Sym( [ 1 .. 3 ] ) gap> H := SymmetricGroup(4); Sym( [ 1 .. 4 ] ) gap> G := DirectProduct(H,K); Group([ (1,2,3,4), (1,2), (5,6,7), (5,6) ]) gap> List( NormalSubgroups(G), Size ); [ 144, 72, 72, 72, 24, 36, 24, 12, 12, 6, 3, 4, 1 ]
$H_k:=\langle\,\left((12k),(1)\right)\,,\,\left((1),(123)\right)\,\rangle \leq S_4\times S_3\,\,,\,\,k=3,4$ are two different Sylow 3-subgroups (order 9) of $\,S_4\times S_3\,$ and, thus, there is not such one normal.
Fact $1$: Suppose $G$ and $H$ are finite groups and $P_G$ and $P_H$ are Sylow $p$-subgroups of $G$ and $H$, respectively. Then $P_G \times P_H$ is a $p$-Sylow subgroup of $G \times H$.
Fact $2$: $A \times B \trianglelefteq G \times H$ if and only if $A \trianglelefteq G$ and $B \trianglelefteq H$.
Fact $3$: If there is at least one Sylow $p$-subgroup that is not normal in $G$, then $G$ has no normal Sylow $p$-subgroup.
Fact $4$: $S_4$ does not have a normal Sylow $3$-subgroup and $S_3$ does not have a normal Sylow $2$-subgroup.
From this you can conclude that $S_4 \times S_3$ does not have a normal Sylow $2$-subgroup or a normal Sylow $3$-subgroup.