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"Abstract Algebra" by Dummit and Foote lacks an exercise or a proof of the classification of (non-abelian) groups of order $8$?

  • On page 581 (third edition) it claims there are only 5 groups of order $8$.

  • The section on $p$-groups classifies groups of order $p^3$ only for $p$ an odd prime.

  • On page 168 it includes a table with the classification of groups of order $8$, but gives no proof.

Between those pages, the book gives classification for groups of several orders ($1,2,3,4,5,6,7,9,10,11,12,13,14,15,\ldots$), but the classification for groups of order $8$ seems to be missing.

It's not even given as an exersice for the reader (or a "check on math.stackexchange" exercise for that matter).

Could it be that the book does include the result somewhere?

If not, is there a one-liner proof (maybe using some of the theorems or exercises in the book itself)? A proof of the style "assume it's not abelian, then take this case and this other case and see it matches the presentation for the dihedral or quaternion group..." is not what I'm looking for, since I think a proof like that would be at least mentioned as an exercise for the reader.

It seems odd that such a thorough, self-contained book does not include this essential classification.

Note I am not asking for a proof of the classification theorem. I am asking about the proof's relation with this specific textbook.

RyeCatcher
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  • May be that is left as an exercise for the reader :-) – Jyrki Lahtonen Jan 12 '24 at 06:25
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    Not a one-liner, but: The abelian groups of order $8$ are easy to dispense with. Any nonabelian group $G$ of order $8$ has an element of order $4$, since any group where every element has order $2$ is abelian (as $aba^{-1}b^{-1} = (ab)^2 = 1$ for all $a, b\in G$), and $G$ is clearly not cyclic. Thus write $G = \langle{a, b\rangle}$ with $a^4 = 1$. But then $\langle{a\rangle}\subset G$ has index $2$ and is thus normal; classify $G$ by the action of $b$ on it. – anomaly Jan 12 '24 at 06:26
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    https://proofwiki.org/wiki/Groups_of_Order_8 – citadel Jan 12 '24 at 06:29
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    Abelian ones are trivial. For non-abelian, it's straightforward from the class equation that the centre has order 2 and the group mod the centre is the Klein group. You can then just brute force the few possibilities almost immediately. – Brevan Ellefsen Jan 12 '24 at 06:31
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    It seems odd that such a thorough, self-contained book does not include this essential classification. I do not agree. There are too many "essential" classifications of finite groups of low order to be carried out in one book. See also Hölder's classification of all groups of order $p^3$ respectively $p^4$, for a prime $p$ (for $p=2$ it includes the groups of order $8$ again). – Dietrich Burde Jan 12 '24 at 10:12
  • exercise is the correct spelling. – Arturo Magidin Jan 12 '24 at 14:55

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