If a group has order $n!$ and has trivial center then it must be isomorphic to $S_n$. Is this statement true?
2026-04-06 14:05:49.1775484349
Group of order $n!$
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This statement is false in general.
Consider the group $S_5 \times S_3$. This group has trivial center and order $5! \cdot 3!=5! \cdot 6=6!$, but it is not isomorphic to $S_6$, because $S_6$ has only the trivial subgroup, $A_6$, and $S_6$ itself as normal subgroups, while $S_5 \times S_3$ has two normal subgroups isomorphic to $S_5$ and $S_3$ respectively.