How many copies of $D_4$ in $S_5$?

Asked Apr 25 '23 at 17:23

Active Apr 26 '23 at 06:27

Viewed 142 times

This question is motivated by studying the Sylow-$p$-subgroups of $S_5$.

From Sylow's theorems, the order of Sylow-2-subgroups has order 8 and are all conjugate to copies of the dihedral groups of $D_4$: $\langle(1 2 3 4), (2 4)\rangle$.

Furthermore, from Sylow's third theorem, the number of conjugates is congruent to 1 (mod 2) and divides 15, and therefore, is 1, 3, 5 or 15.

The answer then is clearly 15 as there are clearly more than 5.

However, I am wondering if anyone can provide a counting argument that the number of copies of $D_4$ in $S_5$ is 15.

Thank you.

edited Apr 26 '23 at 06:27

Maths Rahul

3,293

asked Apr 25 '23 at 17:23

Bond

2

Here's a MathJax tutorial :) – Shaun Apr 25 '23 at 17:45
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You've been told about MathJax before . . . by me. – Shaun Apr 25 '23 at 17:47
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Though true, how do you know in principle that 2-Sylows are dihedral? – Kan't Apr 25 '23 at 17:50
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It may be simpler to start by asking yourself how many copies of $D_4$ are there in $S_4$. Then how many copies of $S_4$ in $S_5$, but some care is needed in combining those two answers. – Jyrki Lahtonen Apr 25 '23 at 17:56
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@orione The reason is this one, for example. They all need to be conjugated to the one with generators $r=(1234)$ and $s=(12)$, which gives the dihedral group of order $8$. – Dietrich Burde Apr 25 '23 at 18:17
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See also Derek's remark here, why there are $15$ $2$-Sylow subgroups in $S_5$. – Dietrich Burde Apr 25 '23 at 18:22
@JyrkiLahtonen I think there are 3 copies of $D_4$ in $S_4$, by the conjugacy classes. And then 5 copies of $S_4$ in $S_5$, by fixing each $i$ when $i \in {1, 2, 3, 4, 5}$. Is this right? – Bond Apr 25 '23 at 18:35
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That is correct. And as you pointed out, the $3\cdot5$ copies resulting from this must be distinct, as they fix a different point (out of five). – Jyrki Lahtonen Apr 25 '23 at 18:39
Why are they "clearly more than 5"? – Kan't Apr 16 '24 at 05:23

How many copies of $D_4$ in $S_5$?

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