show/hide this revision's text 2 minor corrections for better understanding

I want to give some more details and clearifications to the "hierarchy", that has been broached by Carnahan above. The "generic" simple groups are the LIE-TYPE Lie type groups of arbitrary size and the ALTERNATING alternating groups. Furthermore, the main induction step of the classification theorem was, that the centralizer of an involution of a simple group (which an order-2-element/involution is the "only" thing we know have "a-priori" about in an arbitrary simple group by Feit-Thomson!Feit-Thomson) is CLOSE close (!) to simple. So at "low-rank" there is the chance of SPORADIC GROUPS sporadic group branching off from a Lie-Type or alternating group and inductively proceed for some steps until it terminates.

This iterated enlargement inductive process of constructing a much larger simple group from it's involution centralizer being a prescribed (already largegroups ) simple group in extremely rare (!) situations could be thought of some sort of answer to your question. It is by the way one reason for the incredible length of the classification result (a tremendous case-by-case argument)...and for my personal view on the meta-debate above, that sporadics are more sporadic (not more unnatural!) than others, as much to us as to species 8472 ;-) ;-)

Most examples go only one step (and still are very large!), e.g. almost all so-called pariahs:

  • $Ru ON \leftarrow SL_2(4)$SL_3(4)$
  • $ON Ru \leftarrow\;^2SO_5(8)$ "twisted" Lie-type (alike the unitaries over finite fields)
  • $J_4\leftarrow M_{22}\leftarrow$ M_{22}\leftarrow\ldots$ branches off already one induction step beyond Lie (see below)

    • Note that most of these cases already appear as involution centralizers of Lie-type groups, which is somewhat miraculous and was often the reason to study this particular class and find only distinct "irradic" different choices. E.g. $^2G(3^{2n+1})\leftarrow SL_2(3^n)$ and the only other possible case $SL_2(4)\cong SL_2(5)\cong A_5$ lead Janko 1965 to the first new sporadic $J_1$ in almost a century.

    On the other hand there is a VERY remarkable string of induction steps to the Monster, starting and with modifications to the other sporadic groups "involved" in it. It goes roughly as

    $M\leftarrow Co_1 \leftarrow M_{24}\leftarrow SL_3(4)$

    and heavily relies on the already mentioned Golay-Code resp. Steiner System $S(24,8,5)$ - beautiful, very sporadic and purely combinatoric objects! It goes roughly $SL_3(4)\rightarrow M_{24}\rightarrow Co_1\rightarrow M$ or similarly for other groups "involved" in the Monster.Along the induction steps, the combinatorical objects with these groups as automorphism groups can be extended as well(Projective-Plane to Steiner-System to Leech-Lattice to , very roughly like

    Griess-Algebra )$\leftarrow$ Leech-Lattice $\leftarrow$ Steiner-System $\leftarrow$ Projective-Plane

    This is provably impossible for larger numbers (by hard number theory) and is the striking numerical coincidence used in the side-by-side construction of the Golay code, the Steiner System and also the related $24$-dimensional Leech lattice, which is the most dense sphere packing of all dimensions and the reason e.g. the kissing number is known for this dimension!

    show/hide this revision's text 1

    Indeed the question is too vague for a precise answer, but nevertheless somehat natural ;-)

    I want to give some more details and clearifications to the "hierarchy", that has been broached by Carnahan above. The "generic" simple groups are the LIE-TYPE groups of arbitrary size and the ALTERNATING groups. Furthermore, the main induction step of the classification theorem was, that the centralizer of an involution of a simple group (which is the "only" thing we know "a-priori" about an arbitrary simple group by Feit-Thomson!) is CLOSE (!) to simple. So at "low-rank" there is the chance of SPORADIC GROUPS branching off from a Lie-Type group and inductively proceed for some steps until it terminates. This iterated enlargement from already large groups in extremely rare situations could be thought of some sort of answer to your question.

    Most examples go only one step (and still are very large!), e.g. almost all pariahs:

    • $J_1,J_3\leftarrow A_5$
    • $Ly \leftarrow A_{11}$
    • $Ru \leftarrow SL_2(4)$
    • $ON \leftarrow\;^2SO_5(8)$ "twisted" Lie-type (alike the unitaries over finite fields)
    • $J_4\leftarrow M_{22}\leftarrow$ branches off already one induction step beyond Lie (see below)

    On the other hand there is a VERY remarkable string of induction steps to the Monster, starting with the already mentioned Golay-Code resp. Steiner System $S(24,8,5)$ - beautiful, very sporadic and purely combinatoric objects! It goes roughly $SL_3(4)\rightarrow M_{24}\rightarrow Co_1\rightarrow M$ or similarly for other groups "involved" in the Monster.

    Along the induction steps, the combinatorical objects with these groups as automorphism groups can be extended as well (Projective-Plane to Steiner-System to Leech-Lattice to Griess-Algebra)

    One striking numerical reason for this construction and the very exotic behaviour to work exactly for $24$ dimensions (also responsible for the $2^{24}$-factor mentioned above) is:

    $1^2+2^2+\ldots+23^2+24^2=70^2$

    This is provably impossible for larger numbers (by hard number theory) and is the striking numerical coincidence used in the construction of the Golay code, the Steiner System and also the related $24$-dimensional Leech lattice, which is the most dense sphere packing of all dimensions and the reason e.g. the kissing number is known for this dimension!

    Hope that gives some intuition and "personality" for the various sporadics ;-) ;-)