Let $|G|=p^nm$ where $p$ is a prime and $\gcd(p,m)=1$.Suppose that $H$ is a normal subgroup of $G$ of order $p^n$.

Hint: $HK/H \cong K/(H \cap K)$.


Since the claim follows (and is proved along) the standard Sylow theorems presumably we are not allowed to use that.

Hints:

  • Given that $H\unlhd G$ show that $KH=\{kh\mid k\in K, h\in H\}$ is a subgroup of $G$.
  • Show by counting cosets of $H$ inside $KH$ that the order of $KH$ is $|KH|=|H|\cdot [K:H\cap K]$. This is a power of $p$.
  • Apply Lagrange's theorem.

Tags:

Abstract Algebra

Group Theory

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