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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2022; 62(4): 769-772

Published online December 31, 2022

### Group Orders That Imply a Nontrivial p-Core

Rafael Villarroel-Flores

Universidad Autonoma del Estado de Hidalgo, Carretera Pachuca-Tulancingo km. 4. 5, Pachuca 42184 Hgo., Mexico
e-mail : rafaelv@uaeh.edu.mx

Received: October 19, 2021; Revised: January 25, 2022; Accepted: February 9, 2022

### Abstract

Given a prime number p and a natural number m not divisible by p, we propose the problem of finding the smallest number r0 such that for rr0, every group G of order prm has a non-trivial normal p-subgroup. We prove that we can explicitly calculate the number r0 in the case where every group of order prm is solvable for all r, and we obtain the value of r0 for a case where m is a product of two primes.

Keywords: p-core, normal subgroups

### 1. Introduction

Throughout this note, p will be a fixed prime number. We use Op(G) to denote the p-core of G, that is, its largest normal p-subgroup.

We propose the following optimization problem: Given a number m not divisible by p, find the smallest r0 such that every group having order n =prm, with rr0, has a nontrivial p-core Op(G). Denote such number r0 by Λ(p,m). In Theorem 2.1, we will prove that Λ(p,m) is well-defined for any prime p and number m (with pm). In Theorem 2.3 we explicitly determine the value of Λ(p,m) in the case that all groups whose order have the form prm are solvable (for example, if m is prime or if both p and m are odd). Finally, in Section 3, we calculate Λ(2,15), a case that is not covered by the previous theorem.

We remark that the motivation for this research came from the search for examples of finite groups G such that the Brown complex Sp(G) of nontrivial p-subgroups of G (see for example [5] for the definition and properties) is connected but not contractible. It is known that Sp(G) is contractible when G has a nontrivial normal p-subgroup, and Quillen conjectured in [3] that the converse is also true.

### 2. Theorems

Theorem 2.1. For any prime number p and natural number m such that pm, there is a number Λ(p,m) such that if rΛ(p,m), any group of order prm has a non-trivial p-core Op(G).

Proof. Let G be a group of order prm with Op(G)=1. Let P be a Sylow p-subgroup of G. Since the kernel of the action of G on the set of cosets of P is precisely Op(G), we obtain that G embeds in Sm, and so pr divides (m-1)!. Hence, if pr0 is the largest power of p dividing ((m-1)!), we obtain that Λ(p,m)r0+1.

For t,q natural numbers, let γ(t,q) be the product

γ(t,q)=(qt1)(qt11)(q21)(q1),

(note that γ(t,q) can also be defined as (t)!q(q1)!, where (t)!q is the q-factorial of t), and if m=q1t1q2t2qktk is a prime factorization of m, with the qi pairwise distinct and ti>0 for each i, we let Γ(m)=γ(t1,q1)γ(tk,qk). We prove that if ps0 is the largest power of p dividing Γ(m), then Λ(p,m)s0+1.

Theorem 2.2. Let n=psm where pm and s>0. If psΓ(m), then there is a group of order n with Op(G)=1.

Proof. Let K be the group Cq1t1××Cqktk, that is, a product of elementary abelian groups, where m=q1t1qktk and q1,,qk are distinct primes and Cq denotes the cyclic group of order q. Then Γ(m) divides the order of Aut(K), and hence so does ps. Let H be a subgroup of Aut(K) of order ps. For every S∈ H and k∈ K define the map TS,k:KK by TS,k(x)=Sx+k. Then G=TS,kSH,kK is a subgroup of Sym(K). If we identify H with the subgroup of maps of the form TS,0 and K with the subgroup of maps of the form T1K,k, then G is just the semidirect product of K by H. Hence |G|=n. We have that G acts transitively on K in a natural fashion, and the stabilizer of 0∈ K is H, a p-Sylow subgroup of G. Hence the stabilizers of points in K are precisely the Sylow subgroups of G, so their intersection Op(G) contains only the identity KK, as we wanted to prove.

The next theorem will show that the lower bound given by Theorem 2.2 is tight in some cases.

Theorem 2.3. Let n=psm, where pm. If G is a group of order n and ps does not divide Γ(m) then either:

• 1. (Op(G)1), or

• 2. G is not solvable.

Proof. Let G be solvable with order n=psm and Op(G)=1. Let F(G) be the Fitting subgroup of G. Consider the map c:GAutF(G), sending g to cg:F(G)F(G) given by conjugation by g. The restriction of c to P, a p-Sylow subgroup of G, has kernel PCG(F(G)). Since CG(F(G))F(G) (Theorem 7.67 from [4]), and F(G) does not contain elements of order p by our assumption on Op(G), we have P ∩ CG(F(G))=1 and so P acts faithfully on F(G). If m=q1t1qktk is the prime factorization of m, we have that F(G) is the direct product of the Oqi(G) for i=1,,k. Hence PAut(F(G))Aut(Oq1(G))××Aut(Oqk(G)). Let gP such that the action induced by cg on iOq i(G)/Φ(Oq i(G)), is the identity. Since cg acts on each factor Oqi(G)/Φ(Oqi(G)) as the identity, then by Theorem 5.1.4 from [2], we have that it acts as the identity on each Oqi(G). By the faithful action of P on F(G), we have that g=1. This implies that P acts faithfully on iOq i(G)/Φ(Oq i(G)). But then |P| divides the order of the automorphism group of iOq i(G)/Φ(Oq i(G)), which is a product of elementary abelian groups of respective orders qisi with siti for all i. Hence ps=|P| divides Γ(m)

Corollary 2.4. Let ps be the largest power of p that divides Γ(m). If m is prime, or if both p,m are odd, then Λ(p,m)=s+1.

Proof. By Burnside's p,q-theorem, and the Odd Order Theorem, we have that all groups that have order of the form prm for some r are solvable. Therefore, for all r>s, by Theorem 2.3 we have that all groups of order prm have non-trivial p-core.

At this moment, we can prove that in some cases, the group constructed in 2.2 is unique.

Theorem 2.5. Let n=psm where pm and s>0. If psΓ(m), but psΓ(m) for all proper divisors m' of m, then up to isomorphism, the group constructed in the proof of Theorem 2.2 is the only solvable group of order n with Op(G)=1.

Proof. With the notation of the argument of the proof of 2.3, if G is a solvable group of order n with Op(G)=1, we must have that |Oqi(G)|=qiti and Φ(Oqi(G))=1 for all i in order to satisfy the divisibility conditions. Hence Oqi(G) is elementary abelian and a qi-Sylow subgroup for all i, and so G is the semidirect product of a p-Sylow subgroup P of F(G)=Cq1t1××Cqktk with F(G), where the action of P on F(G) by conjugation is faithful. Hence G is isomorphic to the group constructed in the proof of Theorem 2.3.

One case in that we may apply Theorem 2.5 is when n=864. There are 4725 groups of order 864=2533, but only one of them has the property of having a trivial 2-core.

### 3. An Example

An example that cannot be tackled with the previous results is the case p=2, m=35=15. In this case, Γ(15)=(31)(51)=23. Not all groups with order of the form 2r35 are solvable, however, we will prove that Λ(2,15) is actually 4. (The group S5 attests that Λ(2,15)>3.)

Theorem 3.1. Every group G of order 2r35 for r ≥ 4 is such that O2(G)1.

Proof. Let G be a group of order 2r35 for r4. Suppose that O2(G)=1. From Theorem 2.3, we obtain that G is not solvable. We will prove then that O3(G)=1. Suppose otherwise, and let T=O3(G). Then |G/T|=2r5, and so G/T is solvable. Since 2rΓ(5), from Theorem 2.3, we have that O2(G/T)1. Let LG such that O2(G/T)=L/T. Suppose |L/T|=2j. Since O2(G/L)=1, |G/L|=2rj5 and G/L is solvable, we have that 2rj divides Γ(5)=22, that is, rj2. Now, L is also solvable and Γ(3)=31=2, hence if we had j2 we would have O2(L)1, and G would have a non-trivial subnormal 2-subgroup, which contradicts our assumption that O2(G)=1. Hence j=1. But then r12, which contradicts that r4. Hence O3(G)=1. By a similar argument, we get that O5(G)=1.

From [1] we obtain that G is not simple. Hence G has a proper minimal normal subgroup M. From the previous paragraph, we obtain that M is not abelian, since in that case we would have that MF(G). The only possibility is that M=A5. We have then a morphism c:GAut(A5) sending g to cg, the conjugation by g. Since Aut(A5)=S5, and |c(G)|=|Inn(G)||Inn(A5)|=60, in any case the kernel of c is a nontrivial normal 2-subgroup.

### Footnote

This work was partially supported by CONACYT, grant A1-S-45528.

### References

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4. J. S. Rose. A course on group theory. New York: Dover Publications Inc.; 1994.
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