PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 10, October 1997, Pages 2831–2837 S 0002-9939(97)04269-X

COPRIMENESS AMONG IRREDUCIBLE CHARACTER DEGREES OF FINITE SOLVABLE GROUPS

DIANE BENJAMIN

(Communicated by Ronald M. Solomon)

Abstract. Given a finite solvable group G, we say that G has property P_{k} if every set of k distinct irreducible character degrees of G is (setwise) relatively prime. Let k(G) be the smallest positive integer such that G satisfies property P_{k}. We derive a bound, which is quadratic in k(G), for the total number of irreducible character degrees of G. Three exceptional cases occur; examples are constructed which verify the sharpness of the bound in each of these special cases.

1. Introduction

Suppose G is a finite solvable group and let cd(G) denote the set {χ(1) | χ ∈ Irr(G)}. We say that G has property P_{k} if every set of k distinct elements of cd(G) is (setwise) relatively prime. Every finite group G satisfies P_{k} at least for k ≥ | cd(G)|, since 1 ∈ cd(G). The main result of this paper is the following:

Theorem A. Let G be a nonabelian finite solvable group and let k be the smallest positive integer such that G satisfies property P_{k}. Then

| cd(G)| ≤ \begin{cases}3 & if k = 2; \\ 6 & if k = 3; \\ 9 & if k = 4; \\ k^{2} − 3k + 4 & if k ≥ 5. \end{cases}

Following the proof of Theorem A, a collection of examples is presented. In each of the exceptional cases k = 2, 3, 4 the bound is attained. For k = 2, an example is provided by the group SL(2, 3). This group satisfies P_{2} and has 3 irreducible character degrees: cd(SL(2, 3)) = {1, 2, 3}. For k = 3, we construct a group Γ with cd(Γ) = {1, r, s, rs, q^{4}, q^{5}}, where q, r, s are any three primes satisfying q ≡ 3 (mod 4) and q ≡ 1 (mod rs). The group Γ attains the bound in this case. Next, for infinitely many values of k, we construct a group which satisfies property P_{k} and has 3(k − 1) irreducible character degrees. Observe that, for k = 4, such a group satisfies P_{4} and has 9 irreducible character degrees, verifying the sharpness of the bound in this case. It also follows from this infinite set of examples that the best possible bound for | cd(G)| in terms of k cannot be better than the linear bound 3(k − 1).

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors April 4, 1996.

1991 Mathematics Subject Classification. Primary 20C15.

While this result belongs to a genre of problems and results concerning the irreducible character degrees of finite solvable groups (see §2 of [4]), it has a unique flavor. The investigation of property P_{k} was inspired by problem 12.3 of [1] which is, in fact, the k = 2 case of the result. At this point the author would like to express her appreciation to Professor Martin Isaacs for his direction and encouragement in this work, which is a portion of her thesis.

2. Preliminaries

The purpose of this section is to restate facts about the structure and character degrees of a factor group G/K of a finite nonabelian solvable group G with K chosen to be maximal such that G/K remains nonabelian. Notice that, in this situation every proper factor group of G/K is abelian and thus (G/K)′ is the unique minimal normal subgroup of G/K.

(2.1) Lemma. Let G be a finite solvable group and assume that G′ is the unique minimal normal subgroup of G. Then all the nonlinear irreducible characters of G have equal degree f and one of the following situations obtains:

(a) G is a p-group, Z(G) is cyclic and G/Z(G) is elementary abelian of order f^{2}.

(b) G is a Frobenius group with a cyclic Frobenius complement of order f. Also, G′ is the Frobenius kernel and is an elementary abelian p-group.

Proof. This is Lemma 12.3 of [1] with the observation that an abelian Frobenius complement is necessarily cyclic.

(2.2) Theorem. Let K ◁ G be such that G/K is a Frobenius group with kernel N/K, an elementary abelian p-group. Let ψ ∈ Irr(N). Then one of the following holds:

(a) |G : N|ψ(1) ∈ cd(G).

(b) p divides ψ(1).

Proof. This is immediate from Theorem 12.4 of [1].

3. Proof of Theorem A

We begin by proving a key lemma.

(3.1) Lemma. Let G be a finite nonabelian solvable group with G′ ≤ O^{p}(G) for all primes p. Suppose that K ◁ G and K is maximal such that G/K is nonabelian. Then G/K is a Frobenius group with Frobenius kernel N/K, an elementary abelian q-group for some prime q, and a cyclic Frobenius complement. Let f denote the order of the Frobenius complement and assume further that K is chosen so that f is minimal. Then for each linear character λ of N, either λ^{G} is irreducible or λ extends to G. In particular, if χ ∈ Irr(G) lies over a linear character of N, then χ must have degree 1 or f.

It will be handy in the proof to use the standard notation b(G) to denote the largest irreducible character degree of G; that is, the maximum of the set cd(G).

Proof. By hypothesis, if M ◁ G with K < M, then the quotient G/M is abelian. Since G is solvable, it follows that (G/K)′ is the unique minimal normal subgroup of G/K. Now since G has no nonabelian p-factor groups for any prime p, the Frobenius structure of G/K follows from Lemma 2.1 (b).

Fix a linear character λ ∈ Irr(N) and let χ ∈ Irr(G) lie over λ. Set T = I_{G}(λ) and t = |G : T |. Since T/N is cyclic, λ extends to a character λ̂ ∈ Irr(T) and further, by Corollary 6.17 of [1], every element of Irr(T |λ) is an extension of λ. We may then assume without loss of generality that the extension λ̂ is the Clifford correspondent between χ and λ. Thus χ = (λ̂)^{G} and χ_{N} = ∑_{i=1}^{t} λ_{i} , labeled so that λ_{1} = λ. Also note that χ(1) = t.

We are done if t = 1 or t = f, so assume, for a contradiction, that neither happens. In this case 1 < t = χ(1) < f and N < T < G. Let M = kerλ. Since T fixes the linear character λ, it follows that T centralizes N/M and so [T, N] ≤ M. Also [T, N] ◁ G, since both T and N are normal. Let denote quotients mod [T, N]. Then \overline{N} is central in \overline{T} and \overline{T} /\overline{N} is cyclic since it is isomorphic to T/N; thus \overline{T} is abelian. We have \overline{T} is normal and abelian in \overline{G}. By Ito’s Theorem (6.15 of [1]) t = |\overline{G} : \overline{T} | ≥ b(\overline{G}). Also, since kerχ ≥ core_{G}(kerλ) ≥ [T, N], we may view χ as an element of Irr(\overline{G}). We have t = χ(1) ∈ cd(\overline{G}) and thus \overline{G} is nonabelian.

Now, let \overline{G}/\overline{L} be a minimal nonabelian factor of \overline{G}. Clearly the hypothesis on p-factors of G holds for p-factors of \overline{G}. It follows from Lemma 2.1 that \overline{G}/\overline{L} is a Frobenius group with a Frobenius complement of order χ(1) = t < f. Since factors of \overline{G} are factors of G, this contradicts the minimality of f.

It will now be convenient to establish some notation for the proof of Theorem A. For a group G, we define k(G) to be the smallest integer such that G satisfies property P_{k}. Note that k(G) ≤ | cd(G)| and if G is a q-group, for q prime, then equality holds. Given a positive integer q, we define cd_{q}(G) = {n ∈ cd(G) | (q, n) = 1} and cd^{q}(G) = {n ∈ cd(G) | q|n}. If q is prime, then cd(G) is the disjoint union of these two sets. Also, for N ◁G with m ∈ cd(N) if there exists ψ ∈ Irr(N) and χ ∈ Irr(G|ψ) with χ(1) = n and ψ(1) = m, then we will say that n lies over m. Further, for each such m, define a subset of cd(G) by s(m) = {n ∈ cd(G) | n lies over m}. Note that a given n may lie over many different m and each element of s(m) is divisible by m.

To prove Theorem A, we bound each of |cd^{q}(G)| and |cd_{q}(G)| separately in terms of k(G) and add the results. Note that if k = k(G) for a group G, then |cd^{q}(G)| ≤ k − 1 for any positive integer q. On the other hand, given k, examples are available among q-groups, where q is prime, which satisfy | cd(G)| = k. For instance, let Q be the direct product of k − 1 copies of a q-group A having cd(A) = {1, q}. Then cd(Q) = {1, q, q^{2},⋯, q^{k−1}}; thus k = k(Q) and |cd^{q}(Q)| = k − 1. It follows that k − 1 is the best possible bound for |cd^{q}(G)|. Our challenge in proving Theorem A will be to bound |cd_{q}(G)|.

Proof of Theorem A. Let G be a nonabelian finite solvable group and let k = k(G).

Suppose first that G has a nonabelian p-factor group G/K for some prime p. As we have observed, |cd^{p}(G)| ≤ k − 1. Now we consider |cd_{p}(G)|. Fix ψ ∈ Irr(G/K) with ψ(1) = p^{a} > 1. For each character χ ∈ Irr(G) with (p, χ(1)) = 1 we have χ_{K} ∈ Irr(K). By Corollary 6.17 of [1], we have χψ ∈ Irr(G) with degree χψ(1) divisible by p, since χψ(1) = χ(1)p^{a}. This gives an injection from cd_{p}(G) into cd^{p}(G). Thus |cd_{p}(G)| ≤ k − 1 and | cd(G)| ≤ 2(k − 1). In this case the conclusion of the theorem holds. Henceforth we assume that G′ ≤ O^{p}(G) for all primes p.

Now fix K ◁G so that K is maximal with G/K nonabelian. By Lemma 2.1, G/K is a Frobenius group with kernel N/K, an elementary abelian q-group, and with a cyclic complement H/K of order f. Also cd(G/K) = {1, f}. Assume further that K is chosen so that f is minimal. As before, we have |cd^{q}(G)| ≤ k − 1.

To assess |cd_{q}(G)| we will examine how many distinct elements of cd_{q}(G) lie over each element of cd(N). If we write cd(N) = cd^{q}(N) ˙∪ cd_{q}(N), then notice that elements of cd^{q}(G) can lie over only elements of cd^{q}(N), since (q, f) = 1, and elements of cd_{q}(G) lie over only elements of cd_{q}(N). Also, by Theorem 2.2 (a), for each element z ∈ cd_{q}(N) we must have fz ∈ cd(G). This gives an injection from cd_{q}(N) into cd^{f}(G). Again, by hypothesis, |cd^{f}(G)| ≤ k−1 ; thus |cd_{q}(N)| ≤ k−1. It follows that all the elements of cd_{q}(G) lie over the, at most k − 1, elements of cd_{q}(N).

If z ∈ cd_{q}(N), how many elements of cd_{q}(G) can lie over z? By Lemma 3.1, if z = 1, then s(z) = {1, f}. If z > 1, then |s(z)| ≤ k −1, since s(z) ⊆ cd^{z}(G) ≤ k −1, by hypothesis. It follows that |cd_{q}(G)| ≤ 2 + (k − 2)(k − 1) and thus we have:

(∗) | cd(G)| ≤ |cd^{q}(G)| + |cd_{q}(G)| ≤ (k − 1) + 2 + (k − 2)(k − 1) = k^{2} − 2k + 3.

Observe that, when k = 2 the bound (∗) yields | cd(G)| ≤ 3 and when k = 3 the bound (∗) yields | cd(G)| ≤ 6. Thus the first two special cases of Theorem A have been proved. Henceforth we assume that k ≥ 4 and will improve (∗). We continue as before with the Frobenius factor group G/K.

If |cd_{q}(N)| < k − 1, then each of the, at most k − 3, nonlinear character degrees of cd_{q}(N) has at most k − 1 elements of cd_{q}(G) lying over it; thus |cd_{q}(G)| ≤ 2+(k−3)(k−1). This observation along with our bound on cd^{q}(G) yields | cd(G)| ≤ (k − 1) + 2 + (k − 3)(k − 1) = k^{2} − 3k + 4 and there is nothing further to prove in this case.

We may now assume that |cd_{q}(N)| = k − 1. In this case {fx|x ∈ cd_{q}(N)} is a subset of cd(G) of size k − 1. We will show that s(z) ⊆ {z} ∪ {fx|x ∈ cd_{q}(N)} for each z ∈ cd_{q}(N). Recall that an arbitrary member of s(z) has the form rz, where r|f. If rz ∈ s(z) with r > 1, then r divides every member of {rz}∪{fx|x ∈ cd_{q}(N)}. Since the latter set in this union has size k−1, it follows that rz ∈ {fx|x ∈ cd_{q}(N)} and thus we conclude that s(z) ⊆ {z} ∪ {fx|x ∈ cd_{q}(N)} as claimed. It follows that all the members of cd_{q}(G) lie in cd_{q}(N) ∪ {fx|x ∈ cd_{q}(N)}; hence |cd_{q}(G)| ≤ 2(k − 1). Since |cd^{q}(G)| ≤ k − 1, we have | cd(G)| ≤ 3k − 3, in this case.

For k ≥ 4 (and O^{p}(G) = 1), it follows that | cd(G)| is bounded by the maximum of the bounds derived in the two preceding paragraphs. That is,

| cd(G)| ≤ max \begin{cases}(k − 1) + 2 + (k − 3)(k − 1) = k^{2} − 3k + 4, \\ (k − 1) + 2(k − 1) = 3k − 3. \end{cases}

Observe that, for k = 4, the second formula yields a maximum of 9, giving | cd(G)| ≤ 9. In the cases k ≥ 5, the maximum is k^{2} − 3k + 4. Thus Theorem A is proved.

In the next section we give constructions which verify the sharpness of the bound in the exceptional cases k = 3 and k = 4.

4. Constructions

For any three primes q, r, s satisfying q ≡ 3 (mod 4) and q ≡ 1 (mod rs), we construct a group Γ as the semidirect product of a normal Sylow q-subgroup Q and a cyclic group H of order rs such that cd(Γ) = {1, r, s, rs, q^{4}, q^{5}}. The group Γ satisfies P_{3} and has 6 irreducible character degrees; thus providing an example for the sharpness of the bound in the case k = 3. Note that r = 2, s = 3, q = 7 satisfy the conditions.

First we construct Q. Let q be prime with q ≡ 3 (mod 4). Define the group Q of exponent q as follows, where all unspecified commutators are trivial:

Q =⟨x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10} |
[x_{1}, x_{2}] = [x_{3}, x_{4}] = [x_{5}, x_{6}] = [x_{7}, x_{8}] = [x_{9}, x_{10}],
[x_{1}, x_{4}] = [x_{2}, x_{3}] = [x_{5}, x_{8}] = [x_{6}, x_{7}]⟩.

A few observations about the group Q are helpful. First, for notational convenience, label z_{1} = [x_{1}, x_{2}] = [x_{3}, x_{4}] = [x_{5}, x_{6}] = [x_{7}, x_{8}] = [x_{9}, x_{10}] and z_{2} = [x_{1}, x_{4}] = [x_{2}, x_{3}] = [x_{5}, x_{8}] = [x_{6}, x_{7}]. Notice that Z(Q) = Q′ = ⟨z_{1}, z_{2}⟩. From this we see that |Z(Q)| = q^{2} and |Q/Z(Q)| = q^{10}. Thus the group Q is q-special of order q^{12} with exponent q. Further, using additive notation, we may view Q/Z(Q) and Z(Q) as vector spaces over GF (q) with bases \{\overline{x_{1}}, \overline{x_{2}}, ⋯, \overline{x_{10}}\} and \{z_{1}, z_{2}\}, respectively. (Here denotes quotient mod Z(Q).)

What are the degrees of the irreducible characters of Q? Since Q is q-special with |Q : Z(Q)| = q^{10} it follows that Q has q^{10} linear characters. Notice that Q/⟨z_{1}⟩ is isomorphic to the direct product of an extra-special group of order q^{9} having exponent q with Z_{q} × Z_{q}. Also Q/⟨z_{2}⟩ is an extra-special group of order q^{11} having exponent q. These quotients give some information about Irr(Q) and about cd(Q). In particular, we have {1, q^{4}, q^{5}} ⊆ cd(Q). In fact, with the assumption q ≡ 3 mod 4, we can show that these are the only irreducible character degrees of Q. The following fact is required:

Claim. For each nonlinear character θ ∈ Irr(Q) we have (i) Q/ ker(θ) is an extra-special q-group with center Z(θ)/ ker(θ). (ii) Z(θ) ≤ ⟨z_{1}, z_{2}, x_{9}, x_{10}⟩. In particular, we have θ(1) is q^{4} or q^{5} and every automorphism that centralizes ⟨z_{1}, z_{2}, x_{9}, x_{10}⟩ fixes θ.

Fix a nonlinear character θ ∈ Irr(Q). Since Q′ = Z(Q) we have Z(Q) ≰ ker(θ); also Z(Q) · ker(θ) ≤ Z(θ). The nontrivial group Z(θ)/ ker(θ) is cyclic of exponent q; thus it has order q. Further Z(θ) = ker(θ) · Z(Q). It follows that Q/ ker(θ) has a center of order q and that the factor group modulo the center is elementary q-abelian. Thus part (i) holds and from this we have θ(1)^{2} = |Q : Z(θ)|. Now since |⟨z_{1}, z_{2}, x_{9}, x_{10}⟩| = q^{4}, the final statement of the claim will hold once part (ii) is established.

As before, let denote quotient mod Z(Q). Using additive notation in each of the abelian groups \overline{Q} and Z(Q), for an element y ∈ Z(θ) we may write:

\overline{y} = c_{1}\overline{x_{1}} + c_{2}\overline{x_{2}} + ⋯ + c_{10}\overline{x_{10}} with c_{i} ∈ GF (q).

For each element z ∈ Z(Q) and for each generator x_{i} we have [x_{i}, yz] = [x_{i}, y]. Using the defining relations for the group Q, it follows that:

[x_{1}, y] = c_{2}z_{1} + c_{4}z_{2} and [x_{3}, y] = c_{4}z_{1} − c_{2}z_{2}.

Observe that for each element y ∈ Z(θ) and x ∈ Q we have [y, x] ∈ Q′ ∩ ker(θ) = Z(Q) ∩ ker(θ); thus each of [x_{1}, y] and [x_{3}, y] lie in Z(Q) ∩ K. Viewing Z(Q) ∩ K as a 1-dimensional subspace of Z(Q), the vectors c_{2}z_{1} + c_{4}z_{2} and c_{4}z_{1} − c_{2}z_{2} are dependent. If either c_{2} or c_{4} is nonzero, then both are nonzero and c_{4} = αc_{2} and c_{2} = −αc_{4} for some α ∈ GF (q); in which case −α^{2} = 1. Invoking the hypothesis q ≡ 3 mod 4, there is no solution for −α^{2} = 1 in GF (q). This forces c_{2} = c_{4} = 0. Under the same assumption, similar comparisons (e.g. [x_{2}, y] and [x_{4}, y]) force c_{1} = c_{3} = c_{5} = c_{6} = c_{7} = c_{8} = 0. Thus we have Z(θ) ≤ ⟨x_{9}, x_{10}, z_{1}, z_{2}⟩ and the claim is proved.

Now let r and s be distinct primes dividing q − 1. We will define the action of a cyclic group H of order rs on the group Q. First, we define actions on Q by automorphisms a and b of orders r and s respectively. Let δ and ϵ be primitive r and s roots of unity in GF (q) respectively. The actions of a and b are defined on the generators of Q, where all unspecified generators are fixed:

for a : x_{1}^{a} = x_{1}^{δ}, x_{2}^{a} = x_{2}^{δ^{−1}}, x_{3}^{a} = x_{3}^{δ}, x_{4}^{a} = x_{4}^{δ^{−1}};
for b : x_{5}^{b} = x_{5}^{ϵ}, x_{6}^{b} = x_{6}^{ϵ^{−1}}, x_{7}^{b} = x_{7}^{ϵ}, x_{8}^{b} = x_{8}^{ϵ^{−1}}.

We must verify that the proposed definitions interact well with the defining relations among the generators. That is, it must be shown that: if [x_{i}, x_{j}] = [x_{k}, x_{l}], then [x_{i}, x_{j}]^{a} = [x_{k}, x_{l}]^{a} and [x_{i}, x_{j}]^{b} = [x_{k}, x_{l}]^{b}. In fact, more is true. If we write x_{i}^{a} = x_{i}^{α_{i}} and x_{j}^{a} = x_{j}^{α_{j}}, then [x_{i}, x_{j}]^{a} = [x_{i}^{a}, x_{j}^{a}] = [x_{i}^{α_{i}}, x_{j}^{α_{j}}] = [x_{i}, x_{j}]^{α_{i}α_{j}}; and whenever [x_{i}, x_{j}] ≠ 1 we have α_{i}α_{j} = 1. Thus every commutator [x_{i}, x_{j}] is fixed by a. The same holds for b. It follows that a and of b act as automorphisms on Q and that the subgroup ⟨z_{1}, z_{2}, x_{9}, x_{10}⟩ is centralized by both a and b. It is also clear that ab = ba. Since a and b are commuting automorphisms of Q of relatively prime orders, the cyclic group H = ⟨a⟩ × ⟨b⟩ acts on Q. Note that |H| = rs and H centralizes ⟨z_{1}, z_{2}, x_{9}, x_{10}⟩.

Now we consider the action of H on Z(Q) and on Q/Z(Q). As observed, H centralizes Z(Q). To understand the action of H on Q/Z(Q), we return to a vector space point of view. As before, let denote quotients mod Z(Q) and use additive notation. From this perspective, the set \{\overline{x_{1}}, \overline{x_{2}}, …, \overline{x_{10}}\} is a basis for Q/Z(Q) consisting of eigenvectors of a and b with corresponding eigenvalues—

for a : δ, δ^{−1}, δ, δ^{−1}, 1, 1, 1, 1, 1, 1;
for b : 1, 1, 1, 1, ϵ, ϵ^{−1}, ϵ, ϵ^{−1}, 1, 1.

It follows that a and b act diagonally on \overline{Q} and that the orbits of the action of H on \overline{Q} have sizes 1, r, s, and rs.

To complete the construction, define the group: Γ = Q×H.

Now we consider cd(Γ). The coprime action of H on the abelian group Q/Z(Q) is permutation isomorphic to the action of H on Irr(Q/Z(Q)). The orbit sizes of the former action are {1, r, s, rs}. Since H is cyclic, it follows that cd(Γ/Z(Q)) = {1, r, s, rs}. On the other hand, we see from the claim that each nonlinear irreducible character of Q is fixed by H, since H centralizes ⟨z_{1}, z_{2}, x_{9}, x_{10}⟩. Again, since H is cyclic and since the action of H on Q is coprime, the remaining elements of cd(Γ) are exactly the nonlinear irreducible character degrees of cd(Q). It follows that cd(Γ) = {1, r, s, rs, q^{4}, q^{5}}.

The next construction will show that for infinitely many values of k there exist groups which satisfy property P_{k} and have 3(k − 1) irreducible character degrees. It follows that the bound of Theorem A cannot be better than the linear bound 3(k − 1).

For any two distinct primes p and q and for an appropriate n, one can let a cyclic group of order p act on an extra-special q-group of order q^{(2n+1)} such that the irreducible character degrees of the resulting semidirect product are {1, p, q^{n}}. Let the group G be the direct product of m groups of this sort such that all of the primes involved are distinct. Then | cd(G)| = 3^{m}; further, since a single prime divides no more than 3^{(m−1)} irreducible character degrees of G, we have k = k(G) = 3^{(m−1)}+1. It follows that | cd(G)| = 3(k − 1), showing that the bound of Theorem A can be no better than 3(k − 1).

Finally, observe that if, as in the preceding construction, we let Δ = A × B for groups A and B with cd(A) = {1, 2, 3} and cd(B) = {1, 5, 11}, then Δ satisfies P_{4} and has | cd(Δ)| = 9. Thus Δ verifies the sharpness of the bound in the case k = 4 and completes our collection of examples for each of the exceptional cases of Theorem A.

References

[1] I. M. Isaacs, “Character Theory of Finite Groups,” Academic Press, New York, 1976. MR 57:417

[2] B. Huppert, “Endliche Gruppen I,” Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1967. MR 37:302

[3] I. M. Isaacs and D. S. Passman,“A characterization of groups in terms of the degrees of their characters II,” Pacific J. of Math. 24, No.3, (1968) 467-510. MR 39:2864

[4] B. Huppert, “Research in Representation Theory at Mainz (1984 - 1990),” Progress in Mathematics. 95, (1991) 17-36. MR 92c:20011

Department of Mathematics, University of Wisconsin–Platteville, Platteville, Wisconsin 53818

   E-mail address: benjamin@uwplatt.edu