Let $G$ be a group and $H_1$, $H_2$ finite subgroups of $G$. Prove that $|H_1H_2| = |H_1||H_2| /|H_1 \cap H_2| .$
94469
How to prove that $|HK| = \dfrac{|H| \; |K|}{|H \cap K|}$?
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
114590
Prove that any finite group $G$ of even order contains an element of order $2.$
22477
If H is a subgroup of a group G and K is a normal subgroup of G the show that K is a normal subgroup of HK.
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
27215
Intersection of the $p$-sylow and $q$-sylow subgroups of group $G$
70373
Intersection of normal subgroups proof
142938
Sylow 2-subgroups of the group $\mathrm{PSL}(2,q)$
5042
Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
102971
Product of all elements of a finite group with an unique element of order 2
61379
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
155054
$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
127996
$M_1$ and $M_2$ are subgroups and $M_1/N=M_2/N$. Is $M_1\cong M_2$?
77361
How to prove $|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})]$?
39469
If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$?
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.