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What are examples of strikingly anomalous phenomena in mathematics, where just one or a rather small number of cases stand out because they don't fit a general pattern?

This is most interesting when the situation considered is very simple and basic and where the exceptional cases are not merely the lowest-numbered ones: for example, the outer automorphism of the symmetric group $S_{6}$ (which exists for no other $S_{n}$), and the existence of non-standard differentiable structures on $\mathbb{R}^{4}$ (but no other $\mathbb{R}^{n}$).

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    $\begingroup$ It's not exactly like your other examples, but characteristic 2 often behaves differently. $\endgroup$
    – Sam Hopkins
    Commented May 25, 2021 at 17:10
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    $\begingroup$ math.stackexchange.com/q/186103/127263 $\endgroup$
    – Wojowu
    Commented May 25, 2021 at 17:13
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    $\begingroup$ We don't have a "too broad" close reason any more, but this sure seems too broad to me. I like the question, but I think it's not for MO (especially in light of @Wojowu's link), and am voting to close. $\endgroup$
    – LSpice
    Commented May 25, 2021 at 17:19
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    $\begingroup$ I’m voting to close this question because of the reason LSpice mentioned but also the question on Math Stackexchange covers this question. $\endgroup$
    – Sam Hopkins
    Commented May 25, 2021 at 17:20
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    $\begingroup$ @Sam Hopkins often characteristic 2 looks different because of a focus on quadratic-type things. If you look at $p$-power things for a prime $p>2$ then you will find characteristic $p$ behaving in funny ways. $\endgroup$
    – KConrad
    Commented May 25, 2021 at 20:49

3 Answers 3

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The integral of $x^r$ is another power of $x$, for any value of $r$ except $r=-1$, when it's a natural logarithm. That still amazes me.

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    $\begingroup$ Something bothers me a little bit about this, but I'm having trouble articulating it. Here's a clumsy stab at it. $x^r$ has a multiple of another power of $x$ as an anti-derivative. Of course it's just a trivial change of wording, but, from this point of view, we can view $\ln(x)$ as a "generalised multiple" $\lim_{r \to -1} \frac1{r + 1}(x^{r + 1} - 1)$ … not of a power of $x$, but close; and the analogy becomes better when we think about anti-derivatives that vanish at $1$, not (as is somewhat biased against $r = -1$ anyway) of anti-derivatives that vanish (or are undefined) at $0$. $\endgroup$
    – LSpice
    Commented May 26, 2021 at 2:10
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Besides the extra automorphism of $S_6$, both $S_6$ and $S_7$ have exceptional triple covers.

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The special orthogonal group $\operatorname{SO}_8$ is the only $\operatorname{SO}_n$ that has an outer automorphism of order $3$.

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  • $\begingroup$ This seems like an understatement: $\operatorname{SO}_8$, its simply connected cover $\operatorname{Spin}_8$, and its adjoint quotient $\operatorname{PGO}_8$ are the only semisimple matrix groups, orthogonal or otherwise, that have such an automorphism! $\endgroup$
    – LSpice
    Commented May 26, 2021 at 3:00
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    $\begingroup$ Sure $-$ I chose to write just about ${\rm SO}_8$ because that's more generally familiar than spin groups and ${\rm PGO}$ (and thus also didn't make a point of the connection with ${\rm G}_2$). $\endgroup$
    – Noam D. Elkies
    Commented May 26, 2021 at 3:05

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