If $H\leq G$ of index $p$ in $G$ and $p\mid |G|$ then $H\trianglelefteq G$
7108
Normal subgroup of prime index
34567
If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
23008
For all $1 \leq i < j \leq k$, the subtrees $T_i$ and $T_j$ have a vertex in common. Show that $T$ has a vertex which is in all of the $T_i$.
15633
If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
44480
Let $\sigma :\{1,2,3,4,5\} \rightarrow \{1,2,3,4,5\}$ be permutations such that $ \sigma^{-1}(j) \leq \sigma(j)~\forall j, 1 \leq j \leq 5$.
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
67851
Does $|x|^p$ with $0<p<1$ satisfy the triangle inequality on $\mathbb{R}$?
40383
Is it true that $|g| = |\phi(g)|$ for all homomorphisms $\phi: G \to G$ and $g \in G$?
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
156319
Show that a group with $|G| = 33$ contains an element of order $3$
124309
$g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
103806
How to prove triangle inequality for $p$-norm?
106767
$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
52095
Prove either $G=ST$ or |$G|\geq|S|+|T|$
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.