query: How show that not exist two subgroups $H,K \leqslant S_{4} $ such that $H \times K\approx \operatorname{Dih}_4 $
positive_passages.docid: 23077
positive_passages.text: Prove that the dihedral group $D_4$ can not be written as a direct product of two groups
negative_passages.docid: 75670
negative_passages.text: Questions around the number of subgroups of a $p$-group
negative_passages.docid: 28722
negative_passages.text: Prove that $H$ is a abelian subgroup of odd order
negative_passages.docid: 132856
negative_passages.text: Find the derived subgroup of $A_4$
negative_passages.docid: 14611
negative_passages.text: $A_4$ has no subgroup of order $6$?
negative_passages.docid: 99999
negative_passages.text: A question on identifying normal subgroups of $S_4$
negative_passages.docid: 106927
negative_passages.text: How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
negative_passages.docid: 105951
negative_passages.text: Does $G\times K\cong H\times K$ imply $G\cong H$?
negative_passages.docid: 151621
negative_passages.text: $G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
negative_passages.docid: 52105
negative_passages.text: Prove that $HK$ is a subgroup iff $HK=KH$.
negative_passages.docid: 2235
negative_passages.text: Showing $\text{Aut}_\mathsf{Grp}(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}) \cong S_3$
negative_passages.docid: 28181
negative_passages.text: Sylow subgroups of $S_3$ and $S_4$
negative_passages.docid: 111703
negative_passages.text: If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple
negative_passages.docid: 82843
negative_passages.text: Subgroups of $\Bbb{R}^n$ that are closed and discrete
negative_passages.docid: 58033
negative_passages.text: $G\times H\cong G$ with $H$ non-trivial
negative_passages.docid: 16804
negative_passages.text: Prove whether or not a subgroup is normal in the dihedral group $D_4$
negative_passages.docid: 80381
negative_passages.text: Groups with "few" subgroups