Question

Question:If H is a normal subgroup of G and $\left|H\right|\mathbf{=}{\mathit{p}}^{\mathbf{k}}$, show that H is contained in every Sylow p-subgroups of G. may assume Exercise 24 in Section

Short answer

It is proved that H is contained in every Sylow p-subgroups of G.

Step-by-step solution

Second Sylow Theorem

If P and K are Sylow p-subgroups of a group G, then there exists $\mathit{x}\mathbf{\in }\mathit{G}$such that$\mathit{P}\mathbf{=}{\mathit{x}}^{\mathbf{-}\mathbf{1}}\mathit{K}\mathit{x}$ .

Given that H is a normal subgroup of G and$\left|H\right|=p^k$ .

Showing that H is contained in every Sylow p-subgroups of G.

Assume that there is at least one Sylow p-subgroup K of G, such that K is normal to H.

Suppose P is another Sylow p-subgroup such that it is normal to G.

Therefore, bySecond Sylow Theorem, for any arbitrary$x\in G$ ,$P=x^{-1}Kx$ .

Since H is normal of N,$H\in x^{-1}Hx\Rightarrow H=x^{-1}Hx$.

Also, we know H is normal to K,so$H<K$ .

Therefore,$x^{-1}Hx<x^{-1}Kx$ which implies$H<P$ .

Therefore, H is a normal subgroup in P, which is a Sylow p-subgroup.

Hence, it is proved that H is contained in every Sylow p-subgroups of G.