I am really confused about the generalisation of finding out the number of non-abelian groups, as no formula has been introduced yet. Can someone give me some pointers?
2026-04-02 18:42:51.1775155371
How can we find the number of non-abelian groups of order $165$?
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By the Sylow theorems there are exactly $2$ groups of order $165=3\cdot 5\cdot 11$. One of them is the cyclic group $C_{165}\cong C_3\times C_5\times C_{11}$. The second one is the non-abelian group $C_3\times (C_5\ltimes C_{11})$. One starts to show that a group of order $165$ has a normal subgroup of order $11$.
Reference: Written Graduate Qualifying Exam Solutions, University of Maryland 2001 (easy to find online).