Group Theory. Sylow theorems. Group of order 30.

[abhishekkgp]

The following question has two parts to it. I could solve part (a). Need help with part (b).

QUESTION: Let \(\displaystyle G\) be a group of order \(\displaystyle 30\).

a) Show that a 3-sylow subgroup or a 5-sylow subgroup is normal in \(\displaystyle G\).
b) From part (a) show that every 3-sylow subgroup and every 5-sylow subgroup is normal in \(\displaystyle G\).

ATTEMPT:
(a) \(\displaystyle 30=2 \times 3 \times 5\).
The possible values of \(\displaystyle n_5\) are \(\displaystyle 1\) and \(\displaystyle 6\).
The possible values of \(\displaystyle n_3\) are \(\displaystyle 1\) and \(\displaystyle 10\).

Assume no 3-sylow subgroup and no 5-sylow subgroup is normal in \(\displaystyle G\). This means \(\displaystyle n_5=6, n_3=10\).
This gives that there are at least \(\displaystyle (5-1)\times 6 + (3-1) \times 10 + 1= 45\) distinct elements in \(\displaystyle G\). This is clearly a contradiction. Therefore at least one of \(\displaystyle n_3\) and \(\displaystyle n_5\) is \(\displaystyle 1\). Thus this part of the question is solved.

What do i do to solve part (b).

 

[daon2]

If you haven't already, show this: Any group of order 30 has a subgroup of order 15, and the only group of order 15 is Z_15.

This tells you there is phi(15)=8 elements of order 15. If \(\displaystyle n_5=6\) This gives you an additional 24 elements of order 5. Contradiction. Similarly for the other case.

There is probably another way.

 

[abhishekkgp]

If you haven't already, show this: Any group of order 30 has a subgroup of order 15, and the only group of order 15 is Z_15.

This tells you there is phi(15)=8 elements of order 15. If \(\displaystyle n_5=6\) This gives you an additional 24 elements of order 5. Contradiction. Similarly for the other case.

There is probably another way.

Thank you so much!
You are one h-e-l-l of a group theorist!!
How long have you been doing group theory??

 

[daon2]

You are one h-e-l-l of a group theorist!!
How long have you been doing group theory??

These problems aren't exactly easy, but a year from now you will look back at these problems and understand that in some sense they are basic.