# Sylow Subgroups

• May 17th 2009, 01:51 AM
Jason Bourne
Sylow Subgroups
I'm not sure with the following:

(1) What is the order of every Sylow 5-subgroup of $\displaystyle A_5 \times A_5$ ?

(2) Let G and H be arbitrary finite groups. Prove that every sylow p-subgroup of $\displaystyle G \times H$ has the form $\displaystyle P_1 \times Q_1$ where $\displaystyle P_1$ is a sylow p-subgroup of G, and $\displaystyle Q_1$is a sylow p-subgroup of H.

(3) Find the number of sylow 5-subgroups of $\displaystyle A_5 \times A_5$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For (1) i think it's 25, as its's 5.5 ?

For (2) I think I use the Conjugacy sylow theorem somehow?

For (3) I think it might be 26 because $\displaystyle A_5 \times A_5$ is simple?

Thanks! :D
• May 17th 2009, 05:20 AM
TheAbstractionist
Quote:

Originally Posted by Jason Bourne
I'm not sure with the following:

(1) What is the order of every Sylow 5-subgroup of $\displaystyle A_5 \times A_5$ ?

(2) Let G and H be arbitrary finite groups. Prove that every sylow p-subgroup of $\displaystyle G \times H$ has the form $\displaystyle P_1 \times Q_1$ where $\displaystyle P_1$ is a sylow p-subgroup of G, and $\displaystyle Q_1$is a sylow p-subgroup of H.

(3) Find the number of sylow 5-subgroups of $\displaystyle A_5 \times A_5$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For (1) i think it's 25, as its's 5.5 ?

For (2) I think I use the Conjugacy sylow theorem somehow?

For (3) I think it might be 26 because $\displaystyle A_5 \times A_5$ is simple?

Thanks! :D

Hi Jason Bourne.

(2) Determine the maximum power of $\displaystyle p$ dividing the order of $\displaystyle G\times H.$
(3) No, it can’t be 26. The order of $\displaystyle A_5\times A_5$ is $\displaystyle 3600 = 2^4\times3^2\times5^2.$ The number of Sylow 5-subgroups must divide $\displaystyle 2^4\times3^2=144.$ 26 does not divide 144.
for part (3) use part (2) and the fact that $\displaystyle A_5$ has 6 Sylow 5-subgroups. to prove that $\displaystyle P \times Q$ is a Sylow p-subgroup of $\displaystyle G \times H$ if $\displaystyle P$ is a Sylow p-subgroup of $\displaystyle G$ and $\displaystyle Q$ is a Sylow p-subgroup $\displaystyle H$