1. ## Sylows Theorems

Hi got this question where I need to use sylows theorems (or one of them) to determine the numbers of sylow 11-subgroups and sylow 109-subgroups for a group G of order 11990 ( $=2*5*11*109$).

Where do I start with this. Which of sylows theorems do i use and ultimately is it possible to show that G is not simple.

2. Originally Posted by moolimanj
Hi got this question where I need to use sylows theorems (or one of them) to determine the numbers of sylow 11-subgroups and sylow 109-subgroups for a group G of order 11990 ( $=2*5*11*109$).

Where do I start with this. Which of sylows theorems do i use and ultimately is it possible to show that G is not simple.
Let $n$ be number of Sylow $11$-subgroups. Then $n\equiv 1(\bmod 11)$ and $n|11990$. The only possibility is $n=1$. Thus there is only one Sylow subgroup $P$. But $aPa^{-1}$ is also a Sylow subgroup since it is unique it means $P = aPa^{-1}$ thus $P$ is normal.

3. So, do I just show that the group is normal and use the same process for the 109 group?

4. But isn't 2 x 5 x 109 = 1090 $\equiv$ 1 (mod 11)

and

2 x 5 x 11 = 110 $\equiv$ 1 (mod 109)

5. Yeah - thats interesting. Does that imply that there are more than one sylow subgroup for each?