How to prove that if n divides 4! there is a subgroup of order n in S4.

Question

Question as in title.

Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

My approach is rather bullish - I'm simply trying to find subgroups of order n for each factor.

Subgroup of order 1: $<()>$ (Trivial)
Subgroup of order 2: $<(1 2)>$
Subgroup of order 3: $<(1 2 3)>$
Subgroup of order 4: $<(1 2 3 4)>$
Subgroup of order 6: $S_3$ (Symmetric Group on 3 Points)
Subgroup of order 12: $A_4$ (Alternating Group on 4 Points)

But I cannot deduce the others. Is there a more elegant "solution" to this problem. The question cropped up in an introductory group theory class - so I would really appreciate simple answers - I'm familiar with Lagrange's theorem, cyclic groups etc.

Thanks for any help,

Jack

You are only missing 8. What's the symmetry group of the square? — Angina Seng, May 08 '17 at 18:37
@LordSharktheUnknown Ah thank you - D4. Although is there a more elegant way? — Jack, May 08 '17 at 18:39
This is the sort of question where you just have to hack through all cases. — Angina Seng, May 08 '17 at 18:40
Can the Sylow theorems be applied to prove the existence of a subgroup of order $8$, as $8$ and $4!/8$ have no common factor? — avs, May 08 '17 at 18:41

score 0 · Answer 1 · answered May 08 '17 at 19:40

It is well-known that $S_4$ is a Lagrangian group, so that the converse of Lagrange's Theorem is true: for each divisor $d$ of $24$ there is a subgroup of $S_4$ or order $d$. Enumerating all subgroups of $S_4$ "by hand" has been done several times at MSE, e.g., see here, with nice and simple answers.

How to prove that if n divides 4! there is a subgroup of order n in S4.

1 Answers1