Prove $N\cap Z(P)\ne e$ given a $p$-group $P$ and a normal subgroup $N$.
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
90449
Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
51747
Showing that the intersection of all subgroups of order $n$ is normal
121890
Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
70373
Intersection of normal subgroups proof
58205
$ N $ normal in a finite group $ G $, $ |N| = 5 $ and $ |G| $ odd. Why is $ N \subseteq Z(G) $?
147253
Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
80273
Prove the centralizer of an element in group $G$ is a subgroup of $G$
115726
For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$
103423
Normal subgroup where G has order prime
38153
Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .
90332
Prove that that $U(n)$ is an abelian group.
73174
Prove $KN\cong K\times N$ if $K\cap N=\{e\}$
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?