# [seqfan] Re: Subquotients of sporadic simple groups

Sun Sep 20 22:48:15 CEST 2015

```Does the "inverse" of this sequence exist? That is, the number of sporadic simple groups the given group is a subquotient of?

-----Original Message-----
From: Sven Simon <sven-h.simon at gmx.de>
To: 'Sequence Fanatics Discussion list' <seqfan at list.seqfan.eu>
Sent: Sun, Sep 20, 2015 3:29 pm
Subject: [seqfan] Re: Subquotients of sporadic simple groups

As the idea of Charles was interesting, I calculated the values by hand - but
they were double checked, so I am sure, they are correct.
The sequence is

1,2,1,2,1,3,3,1,5,5,2,2,5,5,9,9,6,8,6,2,9,13,6,11,19,20

sequence to the OEIS, using the description of
Charles.
Sven

-----Ursprüngliche Nachricht-----
Von: SeqFan
[mailto:seqfan-bounces at list.seqfan.eu] Im Auftrag von Charles
Greathouse
Gesendet: Donnerstag, 17. September 2015 20:24
An: Sequence Fanatics
Discussion list
Betreff: [seqfan] Subquotients of sporadic simple
groups

Arrange the sporadic simple groups by increasing order 1, 2, ..., 26,
then define a(n) = Number of sporadic simple groups which are subquotients of
the n-th largest sporadic simple group.

I don't think this sequence is in the
OEIS, and it seems interesting. Are all these terms known? Can this sequence be

It's well-known that a(26) = 20, the so-called "happy
family". Trivially
a(1) = 1 and a(2) = 2 since M11 is a subquotient of M12 (and
itself). a(3) = 1 since the Janko group J1 has order 2^3 * ... and 2^4 divides
the orders of M11 and M12.

I've seen diagrams giving (the transitive reduction
of) the subquotient relationships, but I don't know if they're known to be
complete. If not, what's the smallest group with an open question?

Charles
Greathouse
Analyst/Programmer
Case Western Reserve
University

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