Suppose that $H$ and $K$ are two subgroups of a group $G$.Assume that $[G:H]=2$ and $K$ is not a subgroup of $H$.Show that $HK=G$.
132342
Suppose $H$ is a subgroup of $S_n$ but $H$ is not a subgroup of $A_n$. Prove that |$H$ ∩ $A_n$| = $\frac{1}{2}|H|$?
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
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 $G \cong H \times K$
124709
Suppose G be a non-abelian group and H,K be two abelian subgroups of G. Then must HK be an abelian subgroup of G?
79933
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
2761
If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
103002
Group equals union of two subgroups
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
24181
Showing that $G$ is a group under an alternative operation.
13059
If a group is the union of two subgroups, is one subgroup the group itself?
85129
If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
125582
Let $\phi:G_1\to G_2$ be a group homomorphism. Show $\phi(g^{-1})=(\phi(g))^{-1}$.
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$