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Exercise from Herstein:Abstract Algebra.
Please do it without group action as I don't know it. In Herstein there is no mention of Group Action as in the answer given
Show that a group of order $108$ has a normal subgroup of order $9$ or $27$.
Attempt: $108=2^2\times 3^3$. If $n_2$ denotes the number of Sylow $2$ subgroups then $n_2=1+2k| 27\implies n_2=1,3,9,27.$
If $n_3$ denotes the number of Sylow $3$ subgroups then $n_3=1+3k| 8\implies n_3=1,2,4,8.$
If $n_3=1\implies $ the group has a normal subgroup of order $27$ but how to neglect the other choices.
I am confused totally .Please help.