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The number of elements of order 2 in a group is fairly restricted: 0, odd, or infinity. All such possibilities occur already in the trivial group and in dihedral groups.

The number of elements of order 3 in a group can be shown to be similarly restricted: 0, 2 mod 6, or infinity. However, something strange happens: not all possibilities can be realized.

Even worse, there is a fairly small number that I cannot decide whether it is or is not the number of elements of order 3 in a finite group:

Is there a group with exactly 92 elements of order 3?

More boldly, I would like to know (but feel free to answer only the first question):

Exactly which numbers occur as the number of elements of order 3 in a group?

Background: Such questions were studied a bit by Sylow and more heavily by Frobenius. The theorem that the number of elements of order p is equal to −1 mod p is contained in one of Frobenius's 1903 papers. Since elements of order 3 come in pairs, this doubles to give 2 mod 6 for p=3.

However, Frobenius's results were improved some 30 years later by P. Hall who showed that if the Sylow p-subgroups are not cyclic, then the number of elements of order p is −1 mod p2.

If the Sylows are cyclic of order pn, then the number of subgroups of order p is congruent to 1 mod pn by the standard counting method. If the Sylow itself is order p, then the subgroup generated by the elements of order p acts faithfully and transitively on the Sylow subgroups, so for small enough numbers, the subgroup can just be looked up.

In all cases, we can assume the group is finite since the subgroup generated by the elements of a fixed order is finite (assuming there are only finitely many elements of that fixed order).

Easier example: For instance there is no group with exactly 68 elements of order 3, since such a group would have cyclic Sylow 3-subgroups by Hall, order 3 Sylows by the counting, but then would have 34 Sylow 3-subgroups, and so (the subgroup generated by the elements of order 3 would) be a primitive group of degree 34. One checks the list of primitive groups of degree 34 (that is, A34 and S34, both with ginormous Sylow 3-subgroups) to see no such group exists.

One could also try 140, but the action need not be primitive so the table lookup is harder. Such a group has Sylows of order 3, but is not solvable, so is somewhat restricted.

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@Jack Schmidt: May i ask you as to why you chose 92 elements, especially. –  anonymous Aug 13 '10 at 15:21
    
@ Jack Scmidt: This article in the link may be useful for reference springerlink.com/content/4263427343344701 –  anonymous Aug 13 '10 at 15:23
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It is the first number I cannot decide. 2, 8, 14, 20, 26, 32, 38, 44, 50, 56, and 62 are the numbers of elements of order 3 of some group; 68 is not; 74, 80, and 86 are; but 92? I haven't found such a group, but I cannot prove it does not exist. –  Jack Schmidt Aug 13 '10 at 15:24
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@Chandru1: It is a very nice survey of a very interesting area, but it is more asking about uniqueness than existence: are there more than 1 groups with (92 elements of order 3, and so and so many elements of order 5). In computational group theory we use results like this to guess what group we have by looking at the probability distribution of element orders. I do like theorem 2: if the element orders are consecutive integers, then they are a prefix of {1,2,3,4,5,6,7,8}. Theorem 3 looks interesting too: groups with only prime orders are cool, and very restricted. –  Jack Schmidt Aug 13 '10 at 15:43
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Yeah the Deng–Shi paper is very cool too (Theorem 3). –  Jack Schmidt Aug 13 '10 at 15:46

2 Answers 2

up vote 5 down vote accepted

Let p,p' = 2 mod 3 be prime.

Suppose that G is a group with the following properties:

(i) The 3-Sylow subgroup of G is cyclic.

(ii) The number of elements of order 3 in G is 2*p*p'

Claim: There exists a group which _in_addition_ satisfies either:

(iii)(a); G is simple; OR (iii)(b): 3 exactly divides |G|, and G surjects onto Z/3Z.

Remark: Suppose that G is a group with cyclic 3-Sylow subgroups. Since all 3-Sylow subgroups are conjugate, this implies that every subgroup of order 3 in G is also conjugate.

Proof: We may assume that G is not simple, and hence admits a proper normal subgroup H.

Suppose that H is a normal subgroup of G. If 3 divides |H|, then H contains a subgroup of G of order 3. All such subgroups are conjugate in G, and since H is normal, they all lie in H. Thus G and H have the same number of elements of order 3. Moreover, the 3-Sylow of H is clearly cyclic, and so we may replace G by H.

Suppose that (|H|,3) = 1. Then G/H still has a cyclic 3-Sylow subgroup, and hence every element of order 3 in G/H is conjugate. This implies that every element of order 3 in G/H lifts to an element of order 3 in G. Thus G/H has at most 2*p*p' elements of order three. Yet the number of elements of order 3 is (by some theorem of Frobenius?) divisible by 2 and congruent to -1 mod 3. Thus the number of element of G/H of order 3 is either 2 or 2*p*p'. In the latter case, we replace G by G/H.

We may now assume that G/H has exactly 2 elements of order 3. Clearly G/H has a unique subgroup of order 3, which must be normal. Thus G/H and G have a quotient of order |G/H|/3, and thus G contains a normal subgroup F of order 3|H|. As above, we may replace G by F. Note, however, that 3 exactly divides |F|, and F surjects onto Z/3Z, and thus, by Schur-Zassenhaus (overkill in this case, of course), F is a semi-direct product.


My understanding of Jack's argument:

Suppose that p' = 2, so G has 4p elements of order 3. We still assume that the 3-Sylow of G is cyclic. G acts by conjugation on the 2p subgroups of order p, giving a map from G-->S_2p. Let Q be one of these subgroups, and let M be the normalizer of Q. Let P be a 3-Sylow containing Q. Certainly [G:M] = 2p, by the orbit-stabilizer formula. If X is a subgroup of G containing M, then [G:X] = 1,2,p or 2p. Let N be the normalizer of P. Clearly M contains N. Thus P is contained in M, and thus P is a 3-Sylow of M. Similarly, P is also a 3-Sylow subgroup of X. The Sylow theorems applied to M and X thus imply that [X:N] and [M:N] are both 1 mod 3, and thus [X:M] is 1 mod 3. Hence, since [X:M] = 1,2,p or 2p, and, since p = 2 mod 3, either X = M or X = G. Thus M is a maximal subgroup of G, and hence the action of G is primitive. Now we suppose:

Assumption (*): The only primitive subgroups of S_2p are A_2p and S_2p.

Then, we deduce that some quotient of G is either A_2p or S_2p. If G is simple, we deduce that G = A_2p, which doesn't have cyclic 3-Sylows if 2p > 5, and if G is a semi-direct product of Z/3Z with a group of order co-prime to 3, then G cannot surject onto A_2p or S_2p if p > 2. It seems to follow that:

If p = 5,8 mod 9, and Assumption (*) holds, then G cannot have 4p elements of order 3. (The congruence conditions on p ensure that the 3-Sylow of G is cyclic).

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I like this, but it does not handle the case that G has a cyclic Sylow 3-subgroup of order 9. You can handle the order 3 case using the outline for 68: there is no primitive group of degree 46 (that is, Alt(46) and Sym(46)) with exactly 68 elements of order 3. Does your argument adapt to the slightly more general case of 9 exactly divides the order of G? –  Jack Schmidt Aug 15 '10 at 23:10
    
(iiia) Yes, simple G of order exactly divisible by 3 has to be J1, PSL(2,q) with q ≡ {±2,±5} mod 9, or PSLε(3,q) with qε ≡ {2,5} mod 9, where ε = 1 for plain old PSL, and ε = −1 for PSU. –  Jack Schmidt Aug 15 '10 at 23:17
    
I particularly like this style of reduction since I gave up on using the CFSG since I could not determine how the various composition factors interacted. You've shown that (at least for order 3 Sylows), I can assume there is only one factor! –  Jack Schmidt Aug 15 '10 at 23:18
    
This is looking good (5th rev). In particular, it answers the first question and gave me at least 2 new techniques for the second question (which might not even have a reasonable answer). In your reduction, you prove that the number of subgroups of order 3 in G/H is less than the number in G, but I think you use that the former divides the latter. Do you have a reference off hand? If not, it is probably in some textbook treatment of Frobenius's count. –  Jack Schmidt Aug 16 '10 at 5:28
    
Oh, as far as acting primitively: G acts primitively on n points iff G has a maximal subgroup ("M") of index n. It may not act faithfully, so that is the funny business with the quotient. Of course, the Sylow 3-subgroups of a quotient are supposed to be simpler than in the original, so this finishes my attempt in a very clear way. –  Jack Schmidt Aug 16 '10 at 5:45

Here is my attempt, fixed by Septimus Harding:

If H is a group with 92 elements of order 3, then let G be the subgroup generated by those elements. G has finite order by Dicman's lemma, and so G has a Sylow 3-subgroup P.

P must be cyclic, lest Hall's result imply 46≡4 mod 9. Since P is cyclic, 46≡1 mod |P|, so |P| divides 45, so |P| is 1, 3, or 9. 1 is impossible since there would be 0 elements of order 3. 3 is impossible, since then G has 46 Sylow 3-subgroups but no finite group has 46 Sylow p-subgroups. So P is cyclic of order 9.

Now G acts by conjugation on the 46 subgroups of order 3, and since the Sylow 3-subgroups are cyclic and (all of) their order 3-subgroups are conjugate, G acts transitively on the 46 subgroups. Let Q = Ω(P) ≤ P be one of the subgroups of order 3, and let M = NG(Q) be its normalizer. Then N = NG(P) ≤ M and [G:M] = 46. If M ≤ X ≤ G, then both [M:N] and [X:N] are ≡1 mod 3 by Sylow, but [X:N] = [X:M][M:N], so [X:M] ≡ 1 mod 3. However [X:M] divides 46, so [X:M] = 1 or 46, so X = M or G, and so M is a maximal subgroup of G. So G acts primitively on 46 points (the cosets of N, aka, the subgroups of order 3), so G/Core(G,M) is a primitive group of degree 46, that is, G/Core(G,M) is the alternating or the symmetric group on 46 points. Since the Sylow 3-subgroups of the alternating and symmetric groups (on 6 or more points) are not cyclic, we have a contradiction, and no such G exists.


This works to prove that: if n=pq for primes p,q ≡ {5,8} mod 9, and if the only primitive groups of degree pq have non-cyclic Sylow 3-subgroups (like Alt(n) and Sym(n)), then there is no group with exactly 2n elements of order 3, even though 2n ≡ 2 mod 6. I believe primitive groups of degree pq are classified, so this could probably be made completely explicit. 140 elements is not immediately handled, since 70 = 2*5*7, but I suspect the same idea will work.

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