Prove $G=N_G(P)H$ where $H \lhd G$ and $P \in Syl_p(H)$.
9872
Generated Frattini Lemma
11957
G is the product of its Sylow subgroups
151621
$G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
76962
Sylow $p$-subgroups of $GL_n(\mathbb{F}_p)$
98542
Prove that $H$ is a subgroup of an abelian group $G$
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
78263
Prove $GL_n(\Bbb{R})/SL_n(\Bbb{R}) \cong \Bbb{R}^\times$
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
7324
Let $H$ be a subgroup of $G$ with $[G:H]=n$. Let $N$ be the kernel of the left-multiplication action on the cosets of $H$. Show $[G:N]$ divides $n!$
151047
a non trivial example: $G/H$ is isomorphic to $G$
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
100987
Let f be an r-cycle in $S_n$. Given any $h \in S_n$, show that $hfh^{-1} = (h(x_1), h(x_2), \dots, h(x_r))$
40094
Does $G\oplus G \cong H\oplus H$ imply $G\cong H$ in general?
44729
Prove $g$ is a generator if $g^q=1 \pmod p$
85129
If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
144819
Proving $P(n) =n^{\phi(n)} \prod\limits_{d \mid n} \left(\frac{d!}{d^d} \right)^{\mu(n/d)}$
1220
Have $\alpha:G \to H$. If a subgroup $U$ is normal in $H$ prove the pre-image of $U$ is normal in $G$.