Let $G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$ ? ( I know that any subgroup of index smallest prime dividing order of the group would be normal , but this thing is far away from it . Please help . Thanks in advance )
[Math] $G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$
finite-groupsgroup-actionsgroup-theorynormal-subgroups
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Best Answer
Here is a very elementary proof, which uses Lagrange's Theorem, but not Sylow's Theorem.
It is enough to prove that any two subgroups $P,Q$ of order $p$ are equal, because then we must have $gPg^{-1}=P$ for all $g \in G$, so $P$ is normal.
So suppose that $P \ne Q$. Then $P \cap Q = \{1 \}$ by Lagrange. They are both cyclic so $P = \{x^i : 0 \le i < p \}$ and $Q = \{y^i : 0 \le i < p \}$ for some $x,y$.
Since $|G| = pn < p^2$, the elements $x^iy^j$ with $0 \le i,j < p$ cannot all be distinct, so there is an equality $x^iy^j=x^ky^l$ with $(i,j) \ne (k,l)$.
But then $x^{i-k} = y^{l-j}$, contradicting $P \cap Q = \{1 \}$.