Groups - Alternating group question

**Jason Bourne** · May 6th 2009, 09:33 AM

Define $A_n$ as the alternating group. How do I figure out how many 5-cycles $A_5$ has?

**NonCommAlg** · May 6th 2009, 09:45 AM

Originally Posted by Jason Bourne

Define $A_n$ as the alternating group. How do I figure out how many 5-cycles $A_5$ has?

24. suppose $\sigma=(a \ b \ c \ d \ e).$ clearly we have 5! = 120 of these cycles but $\sigma=(b \ c \ d \ e \ a)=(c \ d \ e \ a \ b)=(d \ e \ a \ b \ c)=(e \ a \ b \ c \ d).$ so the number of distinct 5-cycles is 120/5 = 24.

**Jason Bourne** · May 6th 2009, 01:51 PM

Thank you for your post. There are a few more questions concerning this and Sylow theorem that maybe you can help me with.

(i) What is the order of every Sylow 5-subgroup of $A_5$ ?
(ii) How many Sylow 5-subgroups does $A_5$ have?

For (i) I think the answer is 5 because $60=2^{2}.3.5^{1}$
for (ii) I think using one of the Sylow theorems, that the number of Sylow 5-subgroups is of the form $5k+1$ and divides the order of $A_5$ , so I guess from this there is either 1 or 6 Sylow 5-subgroups in $A_5$ , so the answer is 6?? Can anyone verify any of this?

(iii) If $P$ is a Sylow p-subgroup of a finite group $G$ , and if $Q$ is a Sylow p-subgroup of a finite group $H$ , prove that the direct product $P \times Q$ is a Sylow p-subgroup of the direct product $G \times H$ .

For (iii) i attempted to prove it like follows: Take $(p_1,q_1),(p_2,q_2) \in P \times Q$ $(p_1,p_2 \in P, q_1,q_2 \in Q)$

$(p_1,q_1)(p_2^{-1},q_2^{-1}) = (p_1p_2^{-1},q_1q_2^{-1}) \in P \times Q$

Is this the right method? Thanks for any help anyone can provide.

**NonCommAlg** · May 6th 2009, 02:04 PM

Originally Posted by Jason Bourne

Thank you for your post. There are a few more questions concerning this and Sylow theorem that maybe you can help me with.

(i) What is the order of every Sylow 5-subgroup of $A_5$ ?
(ii) How many Sylow 5-subgroups does $A_5$ have?

For (i) I think the answer is 5 because $60=2^{2}.3.5^{1}$
for (ii) I think using one of the Sylow theorems, that the number of Sylow 5-subgroups is of the form $5k+1$ and divides the order of $A_5$ , so I guess from this there is either 1 or 6 Sylow 5-subgroups in $A_5$ , so the answer is 6?? Can anyone verify any of this?

(iii) If $P$ is a Sylow p-subgroup of a finite group $G$ , and if $Q$ is a Sylow p-subgroup of a finite group $H$ , prove that the direct product $P \times Q$ is a Sylow p-subgroup of the direct product $G \times H$ .

For (iii) i attempted to prove it like follows: Take $(p_1,q_1),(p_2,q_2) \in P \times Q$ $(p_1,p_2 \in P, q_1,q_2 \in Q)$

$(p_1,q_1)(p_2^{-1},q_2^{-1}) = (p_1p_2^{-1},q_1q_2^{-1}) \in P \times Q$

Is this the right method? Thanks for any help anyone can provide.

your answers to (i) and (ii) are correct, although you need to explain in (ii) why the number of Sylow 5-subgroups cannot be 1. for that you need to look at your first question about the number

of 5 cycles. for (iii) look at the orders: since P and Q are Sylow p-subgroups we have $|P|=p^m, \ |Q|=p^n, \ |G|=p^m r, \ |H|=p^n s,$ where $\gcd(p,r)=\gcd(p,s)=1.$ then $|P \times Q|=p^{m+n}$ and

$|G \times H|=p^{m+n}rs$ and clearly $\gcd(rs,p)=1.$ this completes the proof.

**TheAbstractionist** · May 7th 2009, 12:34 AM

Originally Posted by Jason Bourne

(ii) I think using one of the Sylow theorems, that the number of Sylow 5-subgroups is of the form $5k+1$ and divides the order of $A_5$ , so I guess from this there is either 1 or 6 Sylow 5-subgroups in $A_5$ , so the answer is 6?? Can anyone verify any of this?

Hi Jason Bourne.

If you are allowed to assume that $A_5$ is simple, then the answer is immediately obvious. If there were just 1 Sylow 5-subgroup, this would be a proper nontrivial normal subgroup of $A_5;$ since $A_5$ is simple, there must therefore be more than 1 Sylow 5-subgroup (indeed more than 1 Sylow subgroup of any order).

**Jason Bourne** · May 7th 2009, 04:11 AM

Originally Posted by TheAbstractionist

If there were just 1 Sylow 5-subgroup, this would be a proper nontrivial normal subgroup of $A_5$

Thanks. How do you know that if there is a unique Sylow 5-subgroup then this is Normal in $A_5$ ?

**TheAbstractionist** · May 7th 2009, 03:01 PM

Originally Posted by Jason Bourne

Thanks. How do you know that if there is a unique Sylow 5-subgroup then this is Normal in $A_5$ ?

Hi Jason Bourne.

Are you aware of the result that for each prime $p$ dividing the order of a finite group $G,$ all Sylow $p$ -subgroups of $G$ are conjugate to each other? It follows that if $G$ has a unique Sylow $p$ -subgroup, then that Sylow $p$ -subgroup is normal in $G.$

Think about it.

**Jason Bourne** · May 8th 2009, 11:35 PM

Originally Posted by TheAbstractionist

Hi Jason Bourne.

Are you aware of the result that for each prime $p$ dividing the order of a finite group $G,$ all Sylow $p$ -subgroups of $G$ are conjugate to each other? It follows that if $G$ has a unique Sylow $p$ -subgroup, then that Sylow $p$ -subgroup is normal in $G.$

Think about it.

Oh yes I just got it! Thank you ,NonCommAlg, TheAbstractionist,thank you Mr Sylow thank you MHF! I don't know why the penny didn't drop before.

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