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For a finite group $G$ one defines the spectrum $\omega(G)$ as the set of element orders of $G$. The set $\omega(G)$ is uniquely determined by the subset $\mu(G)$ consisting of those elements of $\omega(G)$ that are maximal with respect to the divisibility relation.

I would like to know whether for a finite non-abelian simple group $G$ one has $|\mu(G)| \geq 3$.

If this is the case, is there a proof that works for all finite simple groups of Lie type (without treating each family differently)?

Timm von Puttkamer
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  • Did you consider $A_n$, $n\ge 5$? –  Aug 07 '17 at 19:27
  • I did a quick computation with GAP and $\mu(A_n)$ seems to grow with $n$. It seems only $A_5$ and $A_6$ achieve the lower bound with $\mu(A_5) = { 2, 3, 5 }$ and $\mu(A_6) = { 3,4, 5}$. I also remember having checked this for the sporadic groups. – Timm von Puttkamer Aug 07 '17 at 19:39
  • $\mu(A_n)$ should be easy to compute by hand. –  Aug 07 '17 at 19:46
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    It's not clear to me that $\mu(A_n)$ is easy to compute, but showing $\mu(A_n)\ge 3$ is easy. If there are 3 distinct primes in $[n/2,n]$, then $|\mu(A_n)|$ since it contains 3 elements divisible by these elements, which have to be distinct. It seems to hold if $n=13,14$ and $n\ge 17$. It still works if we have three odd prime powers $\le n$ such that the sum of any two of these is $>n$. This holds for $n=7$ and $n\ge 9$. The remaining cases are ok since $2,3,5\in\mu(A_5)$, $3,4,5\in\mu(A_6)$ and $4,5,7\in\mu(A_8)$. – YCor Aug 07 '17 at 20:42

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