Answer to Question #285571 in Abstract Algebra for Aysh

Question #285571

Let G be a group of order 11 2 :1 32 . H ow m any 11 -sylow subgroups and 13 sylow subgroups are there in G ?

1
Expert's answer
2022-01-10T12:26:07-0500

Solution:

"O(G)=11^{2} 13^{2}"

The number of 11 Sylow subgroups are of the form t=1+11 k

since t divides O(G)

"\\Rightarrow t\\ divides\\ 11^{2} 13^{2}\n\n\\\\\\Rightarrow 1+11 k\\ divides\\ 13^{2}\n\n\\\\\\Rightarrow 1+11 k=1\n\n\\\\\\Rightarrow k=0"

Number of 11- Sylow subgroup =1

Since all 11-Sylow subgroups are conjugate and there is only one 11-Sylow subgroup implies the 11-Sylow subgroup is normal.

With the similar argument we can show that there is one 13-Sylow subgroup which is normal.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS