Immersion of Quaternions

Question

Does there exist an immersion of the Quaternion Group in the Symmetric Groups $S_6$ and $S_7$? If it does exist, can you give me an explicit description of that immersion?

Answer 1

The Sylow 2-subgroups of $S_6$ (or $S_7$) are isomorphic to $D_8 \times C_2$ (here $D_8$ is the dihedral group of order 8). Since $Q$ (the quaternion group) is a 2-group, if there is an embedding of $Q$ in $S_6$, then $Q$ would need to be a subgroup of $D_8 \times C_2$. Can you find such a subgroup?

Answer 2

In quaternion group, there are six elements of order $4$ whose squares are equal. Sylow-2 subgroup of $S_6$ and $S_7$ are same (or isomorphic). Therefore, suppose $Q_8$ is embedded in $S_6$. In $S_6$, elements of order $4$ are of the form (or conjugate to) either $(a\, b\, c\, d)(e\,f)$ or $(a\, b\, c\, d)$, their square is $(a\,c)(b\,d)$, and one can see that $(a\,c)(b\,d)$ can be square of $\sigma \in S_6$ if and only if $\sigma$ is of the type $(a\, b\, c\, d)(e\,f)$, $(a\, b\, c\, d)$, $(a\, b\, c\, d)^{-1}(e\,f)^{-1}$ or $(a\, b\, c\, d)^{-1}$; hence we have no six elements of order $4$ in $S_6$ whose squares are equal.