Math Help - 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 ( ).

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.

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 ( ).

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 be number of Sylow -subgroups. Then and . The only possibility is . Thus there is only one Sylow subgroup . But is also a Sylow subgroup since it is unique it means thus is normal.

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

But isn't 2 x 5 x 109 = 1090 1 (mod 11)

and

2 x 5 x 11 = 110 1 (mod 109)

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

PerfectHacker - can you clarify your working/thinking please?