Journal de Théorie des Nombres de Bordeaux

Patrick Meisner

On Incidences of $\varphi $ and $\sigma $ in the Function Field Setting

Jonas Bergström

Neil Dummigan

David Farmer

Sally Koutsoliotas

$\protect \mathrm{GL}_2\times \protect \mathrm{GSp}_2$ $L$-values and Hecke eigenvalue congruences

Dragan Stankov

A necessary and sufficient condition for an algebraic integer to be a Salem number

Jonathan Hickman

James Wright

Counting factorisations of monomials over rings of integers modulo $N$

Stanley Yao Xiao

On binary cubic and quartic forms

Yutaka Konomi

Yutaka KONOMI
On the p-rank of the ideal class group of a normal extension with simple
Galois group
Tome 31, no 3 (2019), p. 671-678.
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Journal de Théorie des Nombres
de Bordeaux 31 (2019), 671–678
On the p-rank of the ideal class group of a normal
extension with simple Galois group
par Yutaka KONOMI
Résumé. Soient p un nombre premier et L une extension normale finie d’un
corps de nombres K dont le groupe de Galois est simple et non abélien. Le
but de cet article est d’estimer la borne inférieure du quotient du p-rang du
groupe de classes d’idéaux de L par le p-rang du groupe de p-classes ambiges
de L par rapport à K.
Abstract. Let p be a prime and L a finite normal extension over a number
field K whose Galois group is simple and non-abelian. The aim of this paper
is to estimate a lower bound of the ratio of the p-rank of the ideal class group
of L to the p-rank of the ambiguous p-class group of L with respect to K.
1. Introduction and the main result
Let p denote a prime number. For an algebraic number field K of finite
degree, denote the p-part of the ideal class group of K by Clp(K) and put
hp(K) = # Clp(K). When L/K is a finite Galois extension, we put
Ambp(L/K) = {x ∈ Clp(L) | ∀ σ ∈ Gal(L/K), xσ = x}
and
ap(L/K) = # Ambp(L/K).
These are called the ambiguous p-class group and the ambiguous p-class
number of L with respect to K, respectively. It is known that
Ambp(L/K) ' Clp(K)
if p - # Gal(L/K). For a finite additive group A, the finite field Fp with
p-elements acts on A/pA. We call dimFp A/pA the p-rank of A and denote
it by p-rank(A).
In the former paper, the author essentially showed the following theorem.
Theorem 1.1 ([2, Theorem 2]). Assume n ≥ 5 and Gal(L/K) is isomor-
phic to An, the alternating group of degree n. Let ln be the maximal prime
number satisfying ln 6= p and ln ≤
√
n. If hp(L) > ap(L/K), then we have
p-rank (Clp(L)/Ambp(L/K)) ≥ ln + 1.
Manuscrit reçu le 16 décembre 2018, révisé le 5 février 2019, accepté le 18 février 2019.
2010 Mathematics Subject Classification. 11R29, 20D06, 20D08.
Mots-clefs. Class group, simple group.
672 Yutaka Konomi
There is ample room for further improvement in Theorem 1.1. In re-
sponse, the aim of this paper is to give explicit lower bounds of the p-rank
of ambiguous ideal class groups on which the simple and non-abelian Galois
group act.
We state the main results. Let G = Gal(L/K) be simple and non-abelian.
Under these notations, we obtain as follows.
Theorem 1.2. Let p denote a prime and set
λ(p) = max{f`(p) | ` is a prime divisor of #G}
where
f`(p) =

min{i ∈ Z>0 | pi ≡ 1 (mod `)}, if p 6= `;
1 +
√
1 + 8 · ordp(#G)
2 , if p = `.
If hp(L) > ap(L/K), we have
p-rank (Clp(L)/Ambp(L/K)) ≥ λ(p).
Theorem 1.3. Fix ` a prime divisor of #G. Let µ` be the maximal value
of the dimensions of elementary abelian `-subgroups of G over F`. For every
prime p different from `, if hp(L) > ap(L/K), then we have
p-rank (Clp(L)/Ambp(L/K)) ≥ µ` + 1.
Theorem 1.4. Let ` be an odd prime divisor of #G, ν` the `-adic order of
the maximal order of an abelian `-subgroup of G. Assume h`(L) > a`(L/K).
Then we have
`-rank (Cl`(L)/Amb`(L/K)) ≥ 2
√
ν`
and
2-rank (Cl2(L)/Amb2(L/K)) ≥ 2
√
µ2.
Remark 1.5 (see Section 3). If one is able to factor #G, it is easy to
calculate λ(p) for each p. In addition, we can also get µ` and ν` by using
a computer. The value λ(p) depends on p, while µ` does not. However,
Theorem 1.2 yields much better numbers than Theorem 1.3 in some cases.
When we write lmax for the maximal prime divisor of #G, λ(p) is less than
or equal to lmax−1. By Dirichlet’s theorem on arithmetic progressions, there
are infinitely many primes p such that λ(p) is equal to not only lmax − 1
but also 1.
Acknowledgments. The author would like to thank the referee for [2],
who communicated to him the ideas of some proofs. He also thanks Yoshi-
chika Iizuka who communicated the proof of Proposition 2.6. The author
is grateful to Professor Shin Nakano for his unfailing encouragement. Last
not least, the author would like to thank the referee of this paper for careful
reading and many valuable suggestions which improved this paper.
On the p-rank of the ideal class group 673
2. Proofs of the main theorems
In this section, we give proofs of the main theorems. When G is a group
and M a G-module, we put
MG = {x ∈M | ∀ g ∈ G, gx = x}.
The following lemma is a key of the proofs of the main Theorems.
Lemma 2.1. Let G be a finite non-abelian simple group, M a G-module
whose order is a p-power. Assume M ) MG and set r = p-rank(M/MG).
Then G embeds into SLr(Fp).
Proof. We show (M/MG)G is trivial. The short exact sequence
0 −→MG −→M −→M/MG −→ 0
produces the exact cohomology sequence
0 −→ (MG)G = MG −→MG −→ (M/MG)G −→ H1(G,MG).
We haveH1(G,MG) = Hom(G,MG) because G acts on triviallyMG. Since
G is simple and non-abelian, Ker(χ) is equal to G for any χ ∈ Hom(G,MG).
Thus, H1(G,MG) is trivial and we obtain (M/MG)G = 0 by this exact
cohomology sequence.
We construct an injective group homomorphism G −→ SLr(Fp). Let
V = {x ∈M/MG | px = 0}. Then, V has the following properties:
(i) The group ring Fp[G] acts on V ,
(ii) Looking at the kernel and cokernel of the group homomorphism
M/MG→M/MG, x 7→ px, we see that #V = #(M/MG)/p(M/MG),
that is, dimFp V = r,
(iii) V 6= 0 and V G = 0.
From (i) and (ii), we get the natural group homomorphism
ρ : G −→ AutFp(V ) ' GLr(Fp).
By the simplicity of G, Ker(ρ) is equal to 1G or G. If Ker(ρ) = G, then
V = V G. This is a contradiction to (iii). Thus, ρ : G −→ GLr(Fp) is
injective.
Similarly, Ker(det ◦ρ : G −→ F×p ) is equal to 1G or G. If Ker(det ◦ρ) =
1G, then G is abelian. This is a contradiction and hence Im(ρ) is subset of
SLr(Fp). 
We give two estimations of lower bounds of r = p-rank(M/MG) in order
to prove Theorem 1.2.
Proposition 2.2. Fix ` a prime divisor of #G. Under the same notations
and assumptions of Lemma 2.1, we have the following estimation of r:
(1) Let p denote a prime different from ` and f the order of p in (Z/`Z)×.
Then r is greater than or equal to f .
674 Yutaka Konomi
(2) If p = `, then r is greater than or equal to 1+
√
1+8·ordp(#G)
2 .
Proof. Note that
# SLr(Fp) = p
r(r−1)
2
r∏
k=2
(pk − 1)
and #G divides # SLr(Fp) by Lemma 2.1.
(1) There exists i such that 1 ≤ i ≤ r and pi ≡ 1 (mod `) from the
assumptions. Hence, we obtain r ≥ f .
(2) This is because ordp(#G) ≤ ordp(# SLr(Fp)) = r(r−1)2 .

By applying Proposition 2.2 to Clp(L)/Ambp(L/K), on which a simple
and non-abelian Galois group acts, we can prove Theorem 1.2.
We calculate another lower bound of r in order to prove Theorem 1.3.
Proposition 2.3. Fix ` a prime divisor of #G. Under the same notations
and assumptions of Lemma 2.1, let B` denote an elementary abelian `-
subgroup of G. Then r is greater than or equal to dimF` B` + 1 for every
prime p different from `.
Proof. Let W` denote the group of `-th roots of unity over Fp. There exists
q such that q is p-power and W` ⊂ Fq. If A ∈ B`, then the eigenvalues of
A are elements in W` because A` = (δij)1≤i,j≤r. Here, δij is the Kronecker
delta. We can regard B` as an elementary abelian `-subgroup of SLr(Fq)
by Lemma 2.1 and noting that Fp ⊂ Fq. Then B` has the following two
properties:
(i) For all X, Y ∈ B`, XY = Y X.
(ii) Since p is different from `, there exist α1, . . . , αr ∈ W` and P ∈
SLr(Fq) such that
P−1AP = (αiδij)1≤i,j≤r.
Consequently, all elements of B` can be diagonalized simultaneously. When
we write Diag(W`) by the group of diagonal matrices whose diagonal el-
ements are included in W`, B` is conjugate to a subgroup of SLr(Fp) ∩
Diag(W`). Hence, dimF` B` ≤ r − 1 holds. 
Applying Proposition 2.3 to Clp(L)/Ambp(L/K) with the simple and
non-abelian Galois group action, we are able to show Theorem 1.3.
We mention a corollary of Theorem 1.3.
Corollary 2.4. Suppose p 6= 2, n ≥ 5 and Gal(L/K) ' An. If hp(L) >
ap(L/K), then
p-rank (Clp(L)/Ambp(L/K)) ≥ 2
⌊
n
4
⌋
+ 1
On the p-rank of the ideal class group 675
holds. Here, b · c means the floor function.
Proof. Put Tk = {(1), (4k − 3 4k − 2)(4k − 1 4k), (4k − 3 4k − 1)(4k −
2 4k), (4k − 3 4k)(4k − 2 4k − 1)} for a positive integer k. Then
⊕k
i=1 Tk
is an elementary 2-subgroup of the alternating group A4k. Hence, µ2 is at
least 2bn4 c. The claim of this corollary follows from Theorem 1.3. 
Remark 2.5. The above corollary is pointed out by the referee for [2] in
the form that
hp(L)/ap(L/K) ≥ p2b
n
4 c+1
if hp(L) > ap(L/K).
To show Theorem 1.4, we give further evaluation of r when p divides
#G.
Proposition 2.6. Under the same notations and assumptions of Lem-
ma 2.1, let ` denote a prime divisor of #G.
(1) The value of r is greater than or equal to 2√µ`.
(2) If ` is an odd, r is greater than or equal to 2√ν`.
Proof. (1). Due to Lemma 2.1, an elementary abelian `-subgroup of G is
embedded in an elementary abelian `-subgroup of SLr(F`). By the Theorem
in [3] and the Sylow theorems, the maximal p-rank of an elementary abelian
p-subgroup of SLr(Fp) is equal to b r
2
4 c for all primes p. Consequently, we
have µ` ≤ r
2
4 .
(2). An abelian `-subgroup of G is embedded in an abelian `-subgroup of
SLr(F`) owing to Lemma 2.1. It is known that the unitriangular matrix
group of degree r over Fp is a Sylow p-subgroup of GLr(Fp) for all primes
p. By the Theorem in [1] and the Sylow theorems, the maximal order of an
abelian p-subgroup of SLr(Fp) is equal to pb
r2
4 c for all odd primes p. If ` is
odd, then we have ν` ≤ r
2
4 . 
Applying Proposition 2.6 to Cl`(L)/Amb`(L/K), we are able to obtain
Theorem 1.4.
3. Examples and tables
In this section, we compute explicit values of λ(p) and µ` for some con-
crete simple groups. To begin with, we confirm that Theorem 1.2 and The-
orem 1.3 are better than Theorem 1.1 for An-extensions.
Example 3.1. With regard to the alternating group A100, we get l100 = 7
and µ2 = 50 by Theorem 1.1 and Corollary 2.4, respectively. Certainly,
Theorem 1.3 is better than Theorem 1.1. We have λ(2) = 82, λ(3) = 88,
676 Yutaka Konomi
λ(5) = λ(7) = 96, λ(11) = 72, λ(101) = 82 and λ(p) ≤ 96 for any prime p.
There are 1229 primes less than 104, and such primes p satisfy that
λ(p) ≤ µ2 + 1 = 51
if and only if
p = 997, 1613, 4021, 4547, 9337, 9781.
Then, λ(997) = λ(1613) = λ(4547) = λ(9781) = 46 and λ(4021) =
λ(9337) = 48 hold. In this case, Theorem 1.2 may be superior to Theo-
rem 1.3.
The classification of finite simple groups is known, that is, every finite
simple group is cyclic, or alternating, or in one of 16 families of groups of
Lie type, or one of 26 sporadic groups. We pick two groups of Lie type,
A2(29) and C3(2), in order to compare the three Theorems.
Example 3.2. We get
#A2(29) =
293
gcd(3, 28)
2∏
i=1
(29i+1 − 1) = 25 · 3 · 5 · 72 · 13 · 293 · 67
and µ2 = µ7 = µ29 = ν7 = ν29 = 2 by using Magma. We see that λ(2) ≥
2√µ2 and λ(`) ≥ 2
√
ν` for ` any odd prime divisor of #G by λ(2) =
λ(7) = λ(13) = 66, λ(3) = λ(5) = 22, λ(67) = 14 and λ(29) = 3. Hence
Theorem 1.2 is superior to Theorem 1.4. Only four primes p = 252589,
304849, 448631, 511211 satisfy that λ(p) < 3 in 78498 primes less than 106.
In this case, Theorem 1.2 is much better than Theorem 1.3 for almost all
primes p.
Example 3.3. We get
#C3(2) = 23
2
3∏
i=1
(22i − 1) = 29 · 34 · 5 · 7,
µ2 = 6, and µ3 = ν3 = 3 by using Magma. If p is odd, then Theorem 1.3
is completely superior to Theorem 1.2 because λ(p) ≤ 6 < µ2 + 1 = 7 for
all odd primes p. If ` is even, Theorem 1.2 is as good as Theorem 1.4, as
λ(2) = 1+
√
73
2 .
For the Lie type groups D4(22), F4(24), E6(3) and Fischer–Griess group
F1, we may not get µ` and ν` even by using a computer. However, we can
On the p-rank of the ideal class group 677
calculate λ(p) because these orders are factored into
#D4(22) = 412(44 − 1)
3∏
i=1
(42i − 1) = 224 · 35 · 54 · 7 · 13 · 172,
#F4(24) = 1624
∏
i∈{2,6,8,12}
(16i − 1)
= 296 · 36 · 54 · 72 · 132 · 174 · 97 · 2412 · 2572 · 673 · 65537,
#E6(3) = 336
∏
i∈{2,5,6,8,9,12}
(3i − 1) = 217 · 336 · 52 · 72 · 112 · 133 · 41 · 73 · 757,
#F1 = 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71.
For example, we get (λ(2), λ(3))=(12, 16), (48, 65536), (756, 12) and (58, 35)
respectively with regard to these four groups.
Lastly, we list the results of calculations using Magma for some other
finite simple groups.
Table 3.1. some Lie Type groups
Group the order λ(2) λ(3) µ2 µ3 ν3
A1(23) 23 · 32 · 7 3 6 3 1 2
A1(27) 27 · 3 · 43 · 127 14 126 7 1 1
A2(33) 24 · 39 · 7 · 132 · 757 756 9 2 6 6
A3(22) 212 · 34 · 52 · 7 · 17 8 16 8 3 3
A4(3) 29 · 310 · 5 · 112 · 13 12 5 4 6 6
B2(24) 216 · 32 · 52 · 172 · 257 16 256 12 2 2
C3(22) 218 · 34 · 53 · 7 · 13 · 17 12 16 12 3 3
G2(3) 26 · 36 · 7 · 13 12 6 3 4 4
G2(22) 212 · 33 · 52 · 7 · 13 12 6 6 2 2
Table 3.2. some sporadic groups
Group the order λ(2) λ(3) µ2 µ3 ν3
M12 26 · 33 · 5 · 11 10 6 3 2 2
J2 27 · 33 · 52 · 7 (1 +
√
57)/2 6 4 2 2
Co3 210 · 37 · 53 · 7 · 11 · 23 11 11 4 5 5
McL 27 · 36 · 53 · 7 · 11 10 6 4 4 4
He 210 · 33 · 52 · 73 · 17 8 16 6 2 2
Suz 213 · 37 · 52 · 7 · 11 · 13 12 6 6 5 5
678 Yutaka Konomi
References
[1] J. T. Goozeff, “Abelian p-subgroups of the general linear group”, J. Aust. Math. Soc. 11
(1970), p. 257-259.
[2] Y. Konomi, “On p-class group of an An-extension”, Proc. Japan Acad. 84 (2008), no. 7,
p. 87-88.
[3] G. N. Thwaites, “The Abelian p-subgroups of GL(n, p) of maximal rank”, Bull. Lond. Math.
Soc. 4 (1972), p. 313-320.
Yutaka Konomi
Department of Mathematics
Meijo University
Tempaku-ku, Nagoya 468-8502, Japan
E-mail: konomi@meijo-u.ac.jp


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