关于16阶群分类的疑问：为何Dummit-Fottee抽象代数书中无16阶非交换群？

Hey there! Great question—this is a common point of confusion with Dummit & Foote's small group summaries, so let’s unpack it step by step.

First, let’s set the record straight: non-abelian order-16 groups absolutely exist—in fact, there are 10 distinct non-abelian isomorphism classes of order-16 groups (compared to 4 abelian ones: (C_{16}), (C_8 \times C_2), (C_4 \times C_4), (C_4 \times C_2 \times C_2)).

So why didn’t you spot them in the 1–20 order group table? Here are the key reasons:

• The table prioritizes brevity for larger p-groups
  16 is (2^4), a higher-power p-group (p=2). Dummit & Foote’s 1–20 summary table typically only lists the most recognizable non-abelian p-groups (like the dihedral group (D_{16}) or quaternion group (Q_{16})) and directs readers to the dedicated finite p-groups chapter for the full classification. The sheer number of order-16 groups (14 total) makes it impractical to list every single one in a compact 1–20 summary.

• Naming conventions might be unfamiliar
  Many order-16 non-abelian groups are defined via semidirect products (e.g., (C_8 \rtimes C_2) in two distinct forms, (C_4 \rtimes C_4) in two forms) or more specialized constructions. The summary table might not spell these out explicitly, instead referencing their structural descriptions that require deeper context from later chapters.

• Check the table’s footnotes or surrounding text
  If you flip back to the text around the small group table, you’ll likely find a note that p-groups with (p^n) where (n \geq 3) have more complex classifications that aren’t fully listed in the summary. The authors save the complete breakdown for sections focused on p-group structure, where they can explain the semidirect products, center subgroups, and automorphism groups that define these non-abelian groups.

To give you a quick taste of the non-abelian order-16 groups you’re missing:

• (D_{16}): The dihedral group of 16 elements (symmetries of an 8-gon)
• (Q_{16}): The generalized quaternion group (a non-abelian group where every element is a power of (a) or (b), with (a^8=1), (b^2=a4), (bab^{-1}=a{-1}))
• Two distinct semidirect products (C_8 \rtimes C_2) (differentiated by the automorphism used to construct the product)
• Two distinct semidirect products (C_4 \rtimes C_4)
• Groups built from (C_4 \times C_2) extended by (C_2) in non-trivial ways

If you want the full classification, head to the chapter on finite p-groups in Dummit & Foote—they’ll walk you through each isomorphism class in detail.

内容的提问来源于stack exchange，提问作者Prince Khan