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Let $^{2}B_{2}(q)$ be Suzuki simple group where $q=2^{2n+1}$. I want to know order of two Suzuki simple groups can divide each other? In other words, suppose that $|^{2}B_{2}(q_{1})|\mid |^{2}B_{2}(q_{2})|$, if this implies $q_{1}=q_{2}$ or not?

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I doubt that this is research level, in particular because the OP doesn't even provide the order formula for the Suzuki groups, which is $q^2(q^2+1)(q-1)$ for $q=2^m$, $m\ge3$ odd. Anyway, for instance if $q_1=2^3$ and $q^2=2^m$ with $m\equiv3\pmod{6}$, then the former Suzuki group order divides the latter one. By easy number theory, one can work out all such cases. – Peter Mueller Dec 25 '12 at 11:55
What is the motivation? If it's just a numerical question, why the tag 'group-theory'? Perhaps the more interesting group-theoretic question is when or if one Suzuki group occurs as a subgroup of another. By construction they all occur in certain $B_2$ Chevalley groups, which in turn sometimes lie in each other (depending on field inclusions). But is there any reason to expect natural inclusions among Suzuki groups? – Jim Humphreys Dec 25 '12 at 13:58
@Jim: Assuming the Suzuki groups are "sufficiently large", any inclusion is natural. In fact, this is true for any (possibly twisted) Lie type. For example, see: MR0734665 (85k:20094) Reviewed Hartley, B.(4-MANC); Shute, G.(1-WIP) Monomorphisms and direct limits of finite groups of Lie type. Quart. J. Math. Oxford Ser. (2) 35 (1984), no. 137, 49–71. – Simon Thomas Dec 25 '12 at 15:37
@Simon: Thanks for calling this article to my attention. Probably our library stores it somewhere in print form, but it isn't available to us online. Judging from Carter's review, it deals with Chevalley and Steinberg (split and quasi-split) types. Does it also treat Suzuki and Ree groups? – Jim Humphreys Dec 25 '12 at 16:48
@Jim: Rutgers doesn't have it online either. My guess/memory is that it also treats the Suzuki and Ree groups as the corresponding result for these groups is also needed for the main application: the classification of the simple periodic linear groups. In any case, these cases were done earlier by Kegal and Stingl. – Simon Thomas Dec 25 '12 at 20:19
up vote 2 down vote accepted

Geoff is on the right track about the way Suzuki groups occur naturally as subgroups of others. But in view of the incomplete formulation of the original question, and the string of comments following (with some references not directly relevant to Suzuki groups), it's worth pointing out sources in the literature.

1) In his 1962 Annals of Mathematics paper, Suzuki was studying special 2-transitive permutation groups (not yet in the Lie context), but already in that paper he worked out explicitly the limited types of possible subgroups which can occur in his new simple groups. These include certain smaller Suzuki groups.

2) In Carter's book Simple Groups of Lie Type (1972) and in Steinberg's 1967-68 Yale lectures on Chevalley groups (the lattter notes available online), the Chevalley groups and twisted groups of types $A, D, E_6$ along with the groups of Suzuki and Ree, were studied in a unified way but with different notational schemes. For more technical treatment, see also Number 3 in the AMS book series The Classification of the Finite Simple Groups by Gorenstein-Lyons-Solomon. Their notation for Suzuki groups (Chapter 2) differs from others. But in all these sources the group involves an odd power $q^2$ of 2, and the group order is given as $q^4 (q^2-1)(q^4+1).$ This convention is not the same as Peter's but is useful for comparisons with a related Chevalley group order. The Suzuki groups themselves are denoted in the Lie theory setting by $^2\!B_2(q)$ or such.

3) The book by G-L-S then formulates Suzuki's subgroup theorem in their Theorem 6.5.4. Here the criterion for one Suzuki group to occur as a subgroup of another one is that its odd exponent of 2 properly divide the odd exponent for the larger group. This is essentially Geoff's observation, confirming the numerical observation made in Peter's comment. For instance, you can have the respective values $q^2=2^3$ and $q^2=2^9$. (There are parallel inclusions of Suzuki groups in Chevalley groups of type $B_2$ and of Chevalley groups in each other, but not such direct inclusions relating all of the groups involved.)

4) The numerical divisibility results are a natural byproduct, but easy to observe directly as in Peter's comment. Whether there are other "accidental" numerical divisibility possibilities for the group orders, I don't know.

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Isn't it the case that the Suzuki group ${\rm Sz}(q)$ occurs as subgroup of ${\rm Sz}(q^{a})$ whenever $a$ is odd, since the former subgroup is the fixed subgroup in the latter group of a field automorphism of order $a?$ In any case, it is clear that the order of the smaller group divides the order of the larger one by an easy number theoretic calculation.

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