A note on 𝑝-parts of conjugacy class sizes of finite groups

Theorem 1.1.

Suppose that G is a finite group and p2 does not divide |xG| for every p-regular element x∈G. Then

Theorem 1.2.

Let G be a finite group, and suppose that p2 does not divide |xG| for every p-regular element x∈G. Then |G⁢:⁢Op⁢(G)|p≤p3 if p≥3.

Lemma 2.1.

Let N be a normal subgroup of a group G.

1. If x∈N, then |xN| divides |xG|.

2. If x∈G, then |(x⁢N)G/N| divides |xG|.

Theorem 2.2.

Let G be a finite solvable group and p a prime. If ap≤p for all a∈clsizepreg⁡(G), then a Sylow p-subgroup of G/Op⁢(G) has order at most p2.

Proof.

This is [9, Theorem 1.1]. ∎

Lemma 2.3.

The prime p does not divide |xG| for any p′-element x∈G if and only if G=Op⁢(G)×Op′⁢(G)

Proof.

This is [1, Lemma 1]. ∎

Lemma 2.4.

Let S be a nonabelian simple group, and let G be an almost simple group with S≤G≤Aut⁡(S). Then the following statements are true.

1. If p is a prime divisor of |S|, then |S|p>|G/S|p.

2. If p is an odd prime divisor of |S| and if ap≤p for all a∈clsizepreg⁡(G), then |G|p=p.

Proof.

(1) is [4, Lemma 3.1], and (2) is [9, Lemma 2.8]. ∎

Lemma 2.5.

Let H be a primitive permutation group on the set Ω={1,…,n}, where n≥2. Assume that p is an odd prime divisor of |H| but ∤⁡p3|H|. Then there exists a proper partition (Ω1,…,Ωd) of Ω such that ⋂i=1dStabH⁡(Ωi) is a p′-group, where 2≤d≤4.

Proof.

This is [7, Lemma 2.1]. ∎

Lemma 2.6.

Suppose that the socle of G is a nonabelian simple group S, and let p be a prime such that p divides |G/S| but not |S|. Then there exist x,y∈S of G-class size divisible by p, where x and y belong to two different S-classes.

Proof.

This is [2, Lemma 3.1]. ∎

Corollary 2.7.

Suppose that the socle of G is a nonabelian simple group S, and let p be a prime such that p divides |G/S| but not |S|. Then there exist x1, x1∈S in different S-classes such that CG⁢(x1) and CG⁢(x2) are p′-groups.

Proof.

Let P∈Sylp⁢(G). Clearly, P≤Out⁡(S). Since S is a p′-group, we know (see, for example, [4, Lemma 2.3]) that P⁢S/S is a normal cyclic subgroup of G/S. Let P1 be a subgroup of P of order p. Then P1⁢S/S is the unique subgroup of G/S with order p. By Lemma 2.6, there exist x1,x2 in S of distinct P1⁢S-class sizes divisible by p. It follows that 𝐂G⁢(x1) and 𝐂G⁢(x2) are p′-groups. ∎

Proposition 2.8.

Let p be an odd prime, and let W be a normal p′-subgroup of G. Assume that p3 does not divide |G/CG⁢(W)| and that W is a direct product of some nonabelian simple groups. Then there exists x∈W such that CG⁢(x)/CG⁢(W) is a p′-group.

Proof.

Clearly, we may assume that p divides |G/𝐂G⁢(W)|. By induction, we may assume 𝐂G⁢(W)=1, and thus p3 does not divide |G|. Assume that W is not minimal normal in G. Since W is a direct product of some minimal normal subgroups of G, we may write W=U1×U2, where U1 and U2 are proper G-invariant subgroups of W. Now we consider quotient groups G/U2 and G/U1. We claim that 𝐂G/U2⁢(W/U2)=𝐂G⁢(U1)/U2 and 𝐂G/U1⁢(W/U1)=𝐂G⁢(U2)/U1. It suffices to show that

It is clear that 𝐂G⁢(U1)/U2≤𝐂G/U2⁢(W/U2). Let g⁢U2∈𝐂G/U2⁢(W/U2). Then [g,U1]≤U2. Since U1 is normal in G, we also have [g,U1]≤U1, and hence [g,U1]≤U1∩U2=1, so g≤𝐂G⁢(U1), as desired. Similarly,

holds. It follows that both G/U2 and G/U1 satisfy the hypothesis.

By induction, there exist x¯⁢U2∈W/U2 and y¯⁢U1∈W/U1 such that

are p′-groups, where x¯,y¯∈W. Let x¯=x1⁢x2 and y¯=y1⁢y2, where xi,yi∈Ui for i=1,2. Then

We claim that 𝐂G/U2⁢(x1⁢U2)=𝐂G⁢(x1)/U2 and 𝐂G/U1⁢(y2⁢U1)=𝐂G⁢(y2)/U1. For example, it is clear that 𝐂G⁢(x1)/U2≤𝐂G/U2⁢(x1⁢U2). On the other hand, if we let L/U2=𝐂G/U2⁢(x1⁢U2), then [L,x1]≤U2∩U1=1 since x1∈U1 and U1 is normal in G. It follows that L≤𝐂G⁢(x1). Hence we see that the claim holds. It follows that 𝐂G⁢(x1)/𝐂G⁢(U1) is a p′-group. Similarly, 𝐂G⁢(y2)/𝐂G⁢(U2) is also a p′-group. Let x=x1⁢y2. Clearly, 𝐂G⁢(x)=𝐂G⁢(x1)∩𝐂G⁢(y2). Since 𝐂G⁢(U1)∩𝐂G⁢(U2)=𝐂G⁢(W)=1, we get that

is a p′-group, as desired.

We now assume that W is minimal normal in G. Moreover, we can assume that G=Op′⁢(G). Otherwise, suppose that Op′⁢(G)<G. Then Op′⁢(G)∩W=W or 1 since W is minimal normal in G. If Op′⁢(G)∩W=1, the result is obvious. If Op′⁢(G)∩W=W, then Op′⁢(G) satisfies the hypothesis by the foregoing arguments, and therefore the result follows.

Then W is a direct product of some isomorphic nonabelian simple groups S1,…,Sn. We may assume n≥2 by Corollary 2.7. Clearly, G is a transitive permutation group on the set {S1,…,Sn}. Let (Δ1,…,Δm) be a system of imprimitivity of G with maximal block size. Then (Δ1,…,Δm) is a partition of {S1,…,Sn}, and all blocks Δi have the same size. Let

Clearly, G is a primitive permutation group on the set Ω with the kernel K, and K is a proper subgroup of G. In particular, all Wi are G-conjugate. Since G=Op′⁢(G), we have |G/K|p=p or p2. By Lemma 2.5, there exists a proper partition (Ω1,…,Ωd) of Ω such that ⋂i=1dStabG/K⁡(Ωi) is a p′-group, where 2≤d≤4.

Applying the inductive hypothesis to K, take wi∈Wi so that 𝐂K⁢(wi)/𝐂K⁢(Wi) is a p′-group, i=1,…,m. Since Wi is nonsolvable, there exist xi,yi,zi∈Wi such that wi, xi, yi and zi have distinct conjugacy class size [3, main theorem]. Clearly, we may choose wi, 1≤i≤m, to be G-conjugate, and we can do the same for xi, yi and zi.

Assume that p divides |𝐂K⁢(W1)|. Note that |K|p≤p by the above arguments. Since all Wi are G-conjugate, p divides |𝐂K⁢(Wi)| for all i. However, the normality of 𝐂K⁢(Wi) in K implies that all Sylow p-subgroups of K are contained in 𝐂K⁢(Wi), where i=1,…,m. It follows that

a contradiction. Hence all 𝐂K⁢(Wi) are p′-groups. This implies that all 𝐂K⁢(xi) are p′-groups. Set

We claim that x fulfills our requirement, and we only consider the case when d=4. Observe that 𝐂K⁢(xi) is a p′ group, and that

because all Wi are normal in K. We conclude that 𝐂K⁢(x) is a p′-group. Since G acts transitively on Ω={Δ1,…,Δm} and wi, xi, yi, zi have different conjugacy class sizes, we have

It follows that 𝐂G⁢(x)⁢K/K≤⋂i=14StabG/K⁡(Ωi) is a p′-group, and so 𝐂G⁢(x) is a p′-group, as desired. ∎

Lemma 2.9.

Let p be an odd prime, and suppose that p2 does not divide |xG| for every p-regular element x∈G. Assume Sol⁢(G)=1; then |G|p≤p.

Proof.

We may assume that p divides |G| and G=Op′⁢(G). Let U be a maximal normal subgroup of G. Then G/U is simple and of order divisible by p, and Lemma 2.4 (2) implies |G/U|p=p. Since |U|p≤p by induction, we conclude that |G|p≤p2. Let W be a minimal normal subgroup. Clearly, W is nonabelian because Sol⁢(G)=1.

Assume that p divides |W|. By the choice of U, we have that W is contained in U, and so |W|p≤|U|p≤p. Therefore, if p divides |W|, then W is a nonabelian simple group with |W|p=p. Since G/(W×𝐂G⁢(W))≲Out⁡(W), by Lemma 2.4 (1), G/(W×𝐂G⁢(W)) is a p′-group. This implies G=W×𝐂G⁢(W) because G=Op′⁢(G). Suppose p is a prime divisor of |𝐂G⁢(W)|. Since

by Lemma 2.3, there exist x1∈Wpreg and x2∈𝐂G⁢(W)preg such that p divides |x1W| and |x2𝐂G⁢(W)|. Then x1×x2∈Gpreg has class size divisible by p2, a contradiction. Hence 𝐂G⁢(W) is a p′-group, and the required result follows.

Assume that p does not divide |W|. Clearly, G>W×𝐂G⁢(W), and p divides |G/(W×𝐂G⁢(W))| because G=Op′⁢(G). Hence, by induction, |𝐂G⁢(W)|p≤p since 𝐂G⁢(W) is a proper normal subgroup of G satisfying the assumption by the observations following Lemma 2.1, and there exists x3∈𝐂G⁢(W)preg such that |x3𝐂G⁢(W)|p=|𝐂G⁢(W)|p by Lemma 2.3. Applying Proposition 2.8, we may take an element x4 of W such that 𝐂G⁢(x4)/𝐂G⁢(W) is a p′-group. Clearly, we have x3×x4∈(W×𝐂G⁢(W))preg, and 𝐂G⁢(x3×x4)=𝐂G⁢(x3)∩𝐂G⁢(x4) because W and 𝐂G⁢(W) are normal in G. Let x=x3×x4. Then, by our hypothesis, we get

Since

and 𝐂G⁢(x4)/𝐂G⁢(W) is a p′-group, we obtain

Proof of Theorem 1.2.

We may assume G=Op′⁢(G) by induction. Since

we get |G/Op⁢(G)|p≤p3. ∎