Let G be an arbitrary group of order 15 How many Slyow 3subg

Let G be an arbitrary group of order 15. How many Slyow 3-subgroups does G have? How many Sylow 5-subgroups does G have? ARe they normal?

Solution

As per Sylow’s 1^st theorem, a finite group G has a p-Sylow subgroup for every prime p.

Also, as per Sylow’s 2^nd theorem, for each prime p, the p-Sylow subgroups of G are conjugate.

As per Sylow’s 3^rd theorem, if |G| = p^k m, where p doesn’t divide m, and if n_p is the number of p-Sylow subgroups of G, then n_p | m and n_p 1 mod p.

Further,the condition n_p = 1 means a p-Sylow subgroup is self-conjugate, i.e., it isa normal subgroup.

Here, |G| = 15 = 3*5 , so that G has one 3-Sylow subgroup and one 5- Sylow subgroup. Further, these two subgroups are normal.