Question

If H is a normal subgroup of G and H is a subgroup of G with $\left|H\right|\mathbf{=}{\mathbf{p}}^{\mathbf{k}}$, prove that H is contained in every sylow p-subgroup of G. [You may assume Exercise 24.]

Short answer

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

Step-by-step solution

Given information

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

Prove H is contained in every Sylow p-subgroup of G

By using exercise 9.4.24, we can say that H is contained in the Sylow p-subgroup of G.

Also,$\left|\mathrm{H}\right|={\mathrm{p}}^{\mathrm{k}}$ therefore by the definition of the Sylow p-subgroup we can say that H is contained in every Sylow p-subgroup of G.

Thus, H is contained in every Sylow p-subgroup of G.