Question

How many Sylow p-subgroup can G possibly have when

P= 5 and$\mathbf{\left|}\mathbf{G}\mathbf{\right|}\mathbf{=}\mathbf{60}$

Short answer

The Sylow 5-subgroups are$\{1,6\}$.

Step-by-step solution

Step-by-Step Solution Step 1: Third Sylow Theorem

The number of Sylow p-subgroups of finite group G divides and$\left|G\right|$ is of the form 1+pk for some non-negative integer k.

Given that p = 5 and $\left|G\right|$= 60.

Finding Sylow 5-subgroups

By Third Sylow Theorem, the number of Sylow 5-subgroups of finite group G, which divides 60 and is the form of 1+5k can be found by using different non-negative values of k.

• Using k as 0 we get 1, which also divides 60.
• Using k as 1 we get 6, which also divides 60.
• Any further values of k will give a result, which will not divide 60.

Therefore, the subgroups are 1 or 6.

Hence, the possible number of Sylow 5-subgroups are $\{1,6\}$.