Question

I think most students who first learn about (finite) groups, eventually learn about the possibility of classifying certain finite groups, and even showing certain finite groups of a given order can't be simple (I'm pretty sure every beginning algebra text has some exercises like this). Up to order 1000, I think there is one that is considered by far the most difficult: 720.

Does anyone know of a proof that there are no simple groups of order 720, which avoids showing, via contradiction, such a group would be $M_{10}$? [To clarify the avoids part, the proof sketched by Derek Holt here, while very nice, would not qualify.]

I should also disqualify the inevitable reference to Burnside's article on this very topic, which I am fairly confident is flawed (or, at the very least, incomplete).

Answer 1

You can only really ask for a proof that avoids a particular fact or construction, if that fact or construction is difficult enough or distinct enough from the thing that you're proving. By that principle, the comments imply that it would be enough to show that Derek Holt's subgroup $\langle a,b,c,e \rangle$ has index 2 in the candidate group $\langle a,b,c,d,e \rangle$. In fact Derek's proof is good: It is easy to show that $\langle a,b,c,e \rangle$ is isomorphic to $L_2(9) = \text{PSL}(2,\mathbb{F}_9)$, which plainly has order 360. ($L_2(9)$ is also isomorphic to $A_6$, but this fact is not needed.) Once you know that, you can also quickly construct $M_{10}$ as well, since you can show that the extra generator $d$ normalizes $L_2(9)$, and that $d^2 \in L_2(9)$.

In that Usenet posting, Derek gave these expressions: $$\begin{matrix} a &=& (2\; 3\; 4)(4\; 6\; 7)(8\; 9\; 10) \\ b &=& (2\;5\;8)(3\;6\;9)(4\;7\;10) \\ c &=& (3\;5\;4\;8)(6\;7\;10\;9) \\ e &=& (1\;2)(5\;8)(6\;7)(9\;10) \end{matrix}.$$ Recall that $\mathbb{F}_9 = (\mathbb{Z}/3)[i]$. You can define a bijection $$\alpha:\mathbb{F}_9 \cup \{\infty\} \to \{1,2,\ldots,10\}$$ by the formula $$\alpha(x+iy) = 2+x+3y \qquad \alpha(\infty) = 1,$$ using the gauche embedding $\mathbb{Z}/3 = \{0,1,2\} \subseteq \mathbb{Z}$. Then it is easy to check these expressions (using more normal arithmetic in $\mathbb{F}_9$): $$a(z) = z+1 \qquad b(z) = z+i \qquad c(z) = iz \qquad e(z) = 1/z.$$ So, that's $L_2(9)$.

Answer 2

How about if we follow Derek Holt's beautiful argument until we establish the following two facts?

1) G has 10 Sylow 3-subgroups.

2) Let P_3 be a Sylow 3-subgroup of G. In the action of G on Syl_3(G), every nonidentity element of P_3 has cycle type (3,3,3,1).

Since G is simple, the action on Syl_3(G) embeds G in A_10. In A_10, no element of type (3,3,3,1) commutes with an element of order two. Thus if Q is a Sylow 2-subgroup of N_G(P_3), no nonidentity element of Q centralizes any nonidentity element of P_3. Since |Q|=8, it follows that all eight nonidentity elements of P_3 are conjugate in N_G(P_3). Therefore, G contains exactly one conjugacy class of elements of order three. Let g_3 be an element of this class. No element of order 5 in A_10 commutes with g_3, and it follows now that |C_G(g_3)|=9, so G has 80 elements of order 3.

Since 5 does not divide |N_G(P_3)|, we see that every element of order 5 in G has cycle type (5,5) in the action on Syl_3(G). Using Sylow's Theorem (and the fact that G is not S_6), we see that G has either 16 or 36 Sylow 5-subgroups. Let P_5 be a Sylow 5-subgroup of G. By Burnside transfer, we cannot have |N_G(P_5)|=45. Therefore, |N_G(P_5)|=20. No element of order two in A_10 commutes with an element of cycle type (5,5). Therefore, N_G(P_5) must induce all of Aut(P_5), so all nonidentity elements of P_5 are conjugate in N_G(P_5) and therefore, G has one conjugacy class of elements of order 5. There are 144 such elements. Let g_5 be one such element.

Now let us consider the character table of G. By the arguments above, each of g_3 and g_5 are conjugate to all of their nonidentity powers, and it follows that all irreducible characters of G take integer values on g_3 and g_5. Moreover, these integers have absolute value at most two, as can be seen using the orthonormality conditions on the character table and the sizes of the given conjugacy classes.

Now N_G(P_3) has one more conjugacy class than Q (as defined above). It follows from this that the irreducible characters of N_G(P_3) are those with kernel containing P_3 and one more, call it Y. Using orthonormality conditions on the character table, we get Y(1)=8, Y(g_3)=-1 and Y(q)=0 for all q not of order 1 or 3. Induce Y up to G to get a character Z that takes the value 80 on 1, -1 on the class of g_3 and 0 elsewhere. For any irreducible character X, the inner product of Z and X is

80(X(1)-X(g_3))/720.

It follows that X(1)-X(g_3) is divisible by 9.

Now N_G(P_5)=Z_5.Z_4 has a character A such that A(1)=4, A(g_5)=-1 and A(q)=0 if q does not have order 1 or 5. Induce A up to G to get B. Arguing as we did with Z, we see that for every irreducible character X, X(1)-X(g_5) is divisible by 5.

Now using basic facts about irreducible characters, we see that for any irreducible character X of G, the triple (X(1),X(g_3),X(g_5)) is one of

(1,1,1),(8,-1,-2),(9,0,-1),(10,1,0),(16,-2,1),(18,0,-2) or (20,2,0).

Any class function X satisfying X(1)=18 and X(g_5)=-2 has norm larger than 1. If X is a class function of norm 1 satisfying X(1)=20 and X(g_3)=2 then X(q)=0 for all q not of order 1 or 3. But then the inner product of X and the trivial character is positive. Similarly, if X has norm 1 and (X(1),X(g_3),X(g_5))=(8,-1,-2) then X is zero on all remaining classes and the inner product of X and the trivial character is negative. We see now that all nontrivial irreducible characters of G have degree 9,10, or 16. We get a contradiction when trying to add up squares of these degrees to get 719.

This proof has a clear disadvantage of greater length when compared with the argument of Holt, as clarified beautifully by Greg Kuperberg. However, it has the advantage of alerting us to the important fact that, given a purported simple group G, it can be profitable to consider large subgroups of G whose characters we understand, in particular those that are Frobenius groups.