Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
add comment

3 Answers

up vote 5 down vote accepted

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.

share|improve this answer
add comment

$$H_k:=\langle\,\left((12k),(1)\right)\,,\,\left((1),(123)\right)\,\rangle \leq S_4\times S_3\,\,,\,\,k=3,4$$ are two different Sylow 3-subgroups (order 9) of $\,S_4\times S_3\,$ and, thus, there is not such one normal.

share|improve this answer
add comment

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 ]
share|improve this answer
    
I used a List() command to shrink the code, and added a little explanation to what the calculation shows. @Tim Duff: Feel free to revert if you don't like it. –  Jack Schmidt Jul 7 '12 at 13:48
    
Fancy - I like it. Guess it's time to learn more GAP... –  Tim Duff Jul 8 '12 at 0:00
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.