EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 5, 2017, 955-966
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

The Group of Units of Integral Group Rings of
Extra-Special 2-Groups

Cristiane de Mello

Mathematics Institute, Federal University of the State of Rio de Janeiro, Rio de Janeiro,
Brazil

Abstract. One of the main problems on group rings is to determine its group of units. In this
paper, we describe the group of units of integral group rings of two extra-special 2-groups: one of
order 32, the central product of two copies of D4, and another of order 128, the central product of
three copies of D4.

2010 Mathematics Subject Classifications: 16S34, 16U60, 20D15

Key Words and Phrases: Group rings, Units in integral group rings, The group of units of
ZD4, The group of units of ZG, G extra-special 2-group.

1. Introduction

A important problem on group rings is to describe precisely the group of units U(RG)
of a group ring RG, where R is a commutative ring with identity and G is a finite group.
The high degree of complexity of this problem became evident in the seventies, when
it was proven that, in general, the group of units contains a non abelian free subgroup.
Many researchers, using techniques of group representation theory and algebraic number
theory, presented an explicit description of the group of units, a description of the general
structure of U(RG) or a set of generators of a finite index subgroup of U(RG). On these
subjects we could quote A. K. Bhandari and I. S. Luthar [1], A. Bovdi and F. C. Polcino
Milies [2], R. A. Ferraz [3], A. Giambruno and S. K. Seghal [4], E. G. Goodaire and E.
Jespers [5], E. Jespers and G. Leal [8], [9], E. Jespers and G. Leal and F. C. Polcino Milies
[10], E. Jespers and M. M. Parmenter and S. K. Sehgal [11], F. C. Polcino Milies [12],
J. Ritter and S. K. Sehgal [13], [14], [15] and two new books by E. Jespers and A. del
Rio [6], [7], and many others. In [6, 7], for many finite groups G, methods are given to
describe all the rational representations of G. In particular, for nilpotent nite groups G
the Wedderburn decomposition of the rational group algebra QG is explicitly given via the
construction of a complete set of matrix units of QG. From this one obtains an explicit
set of nitely many generators for a subgroup of nite index of the unit group U(ZG); these

Email addresses: demello.cristiane@gmail.com (C. Mello)

http://www.ejpam.com 955 c© 2017 EJPAM All rights reserved.



C. Mello / Eur. J. Pure Appl. Math, 10 (5) (2017), 955-966 956

generators are the so called Bass units and bicyclic units. Actually, it known that for
many nite groupsG the Bass units together with the bicyclic units generate a subgroup
of nite index, the groups excluded are determined by the existence of exceptional simple
components of QG (such as non-commutative division algebras and M2(Q)). In general
it remains a problem to describe the full unit group U(ZG). In [6] this has been done for
several examples of nite groups.

F. C. Polcino Milies [10] was the first to describe the unit group U(ZD4), where D4

denotes the dihedral group of order 8. Later, E. Jespers and G. Leal [6] described the same
group using a different method that also was applied to other 2-groups. In this paper we
describe∗ the group of units of the integral group rings of two extra-special 2-groups: G1

of order 32, the central product of two copies of D4, and G2 of order 128, the central
product of three copies of D4.

2. Notation and Terminology

Before we begin, we will recall the definition of extra-special p-group, where p is a
prime number:

Definition 1. A p-group G is called extra-special if it is not abelian and its commutator
subgroup G′ coincides with its center Z(G) and is of order p. In particular, every extra-
special p-group is the central product of non-abelian subgroups of order p3.

Let D4 =
〈
b, v| b2 = v4 = 1 and bvbv = 1

〉
be the dihedral group of order 8. We also

adopt the following notation for elements of D4: a = bv2, s = v2, t = bv, u = vb, w = v3.
Let D , D1 and D2 be groups isomorphic D4, where the indices used are necessary to

differentiate the elements. For example, we denote by bi and vi, 1 ≤ i ≤ 2, the elements of
D i wich correspond respectively to b and v in D . The distinction of the elements is essential
for the proofs we make in this work. With this notation, si = v2i , 1 ≤ i ≤ 2, and the
correspondence of the other elements is obvious. Thus we have the extra-special 2-group
G1 of order 32, the central product of two copies of D4, and the 2-group extra-special G2

of order 128, the central product of three copies of D4:

G1 = D ×D1
/
{1, ss1}, G2 = D ×D1 ×D2

/
{1, ss1, ss2, s1s2}.

The elements of G1/G
′
1, where G′1 = {1, s} is the commutator subgroup of G1, and the

elements of G2/G
′
2, where G′2 = {1, s} is the commutator subgroup of G2, are denoted as

follows:

G1/G
′
1 =

{
1, b, u, v, b1, u1, v1, bb1, bu1, bv1, ub1, uu1, uv1, vb1, vu1, vv1

}
;

∗The calculations given are complete and are independent of the general frame work given in [1, 2].



C. Mello / Eur. J. Pure Appl. Math, 10 (5) (2017), 955-966 957

G2/G
′
2 =



1, b, u, v, b1, u1, v1, b2, u2, v2, bb1, bu1, bv1, bb2, bu2, bv2,

ub1, uu1, uv1, ub2, uu2, uv2, vb1, vu1, vv1, vb2, vu2, vv2,

b1b2, b1u2, b1v2, u1b2, u1u2, u1v2, v1b2, v1u2, v1v2, bb1b2,

bb1u2, bb1v2, bu1b2, bu1u2, bu1v2, bv1b2, bv1u2, bv1v2, ub1b2,

ub1u2, ub1v2, uu1b2, uu1u2, uu1v2, uv1b2, uv1u2, uv1v2,

vb1b2, vb1u2, vb1v2, vu1b2, vu1u2, vu1v2, vv1b2, vv1u2, vv1v2


.

In this paper we describe U(ZG1) and U(ZG2), the group of units of ZG1 and ZG2,
respectively. To this end, we still need to fixe more notations:

(i) If G is a finite extra-special 2-group and G′ = {1, s} is its commutator subgroup,
then U2 denotes the subgroup of U(ZG) defined by

U2 = U(ZG) ∩
(
QG
(1− s

2

)
+
(1 + s

2

))
.

(ii) Let R be a domain and GLn(R) the group of invertible n by n matrices with coe-
ficients in R. If S is a subset of GLn(R), then Sdet=1 denotes a set of matrix units
of S with determinant 1. Similarly, Sdet=±1 denotes a set of matrix units of S with
determinant ±1. By In we denote the identity matrix in GLn(R), and if Sdet=1 is a
multiplicative group, then

S det=1 = Sdet=1/({In,−In} ∩ Sdet=1).

Similarly,
S det=±1 = Sdet=±1/({In,−In} ∩ Sdet=±1).

3. Auxiliary Results

For the main results, we will need the following:

Proposition 1. [6, Lemma 4.2] The group ring QD4 admits the following decomposition:

QD4 = QD4

(1 + s

2

)
⊕QD4

(1− s
2

)
,

where QD4

(1 + s

2

)
∼= Q4 and QD4

(1− s
2

)
∼= M2(Q).

For the sequence of this work, we will fix the following representation of D4 on M2(Q):

• b 7→ e11 − e22 =

[
1 0
0 −1

]
;

• v 7→ −e12 + e21 =

[
0 −1
1 0

]
.

Proposition 2. [6, P roposition 3.2] Writing E =
1− s

2
and using the above fixed repre-

sentation, an elementary Q-basis matrix of QD4

(1− s
2

)
∼= M2(Q) is:



C. Mello / Eur. J. Pure Appl. Math, 10 (5) (2017), 955-966 958

e11 =
(1 + b

2

)
E, e12 =

(vb− v

2

)
E,

e21 =
(v + vb

2

)
E, e22 =

(1− b
2

)
E.

Proposition 3. [6, P roposition 2.4] Let G be an finite extra-special 2-group, G′ = {1, s}
the commutator subgroup of G and ε the augmentation mapping on QG. Then:

(i) U2 = {u = 1 + α(1− s)|u ∈ U(ZG), α ∈ ZG};

(ii) If V = {u = 1 + α (1− s)|u ∈ U(ZG), α ∈ ZG and ε(α) is even}, then V ∼= U2/G
′;

(iii) If G/G′ has exponent at most 4, then U(ZG) = ±GV .

Theorem 1. [6, Theorem 4.3] U(ZD4) = ±D4V and V is isomorphic to the group of
2-by-2 matrices [

2Z + 1 4Z
2Z 2Z + 1

]
det=1

.

Proposition 4. Let G be an finite extra-special 2-group and G′ = {1, s} its commutator

subgroup. Se Ĝ′ =
1 + s

2
, then the component QG(1− Ĝ′) = QG

(1− s
2

)
in the decompo-

sition of QG is simple.

Proof. Let E =
1− s

2
and e be a non-trivial central idempotent of QG(E). If Z(G)

denotes the center of G and Cg denotes the class of conjugation of g ∈ G, then

e =
∑

g∈Z(G)

αgg +
∑

g/∈Z(G)

αgCg,

where Cg =
∑
x∈Cg

x. Since G ′ = {1, s}, if g is not central, then Cg = {g, gs}. Consequently,

e =
∑

g∈Z(G)

αgg +
∑

g/∈Z(G)

αg(g + gs) =
∑

g∈Z(G)

αgg + (1 + s)
∑

g/∈Z(G)

αgg.

Since e is an idempotent of QG(E), it follows eE = e. Hence, as (1 + s)E = 0,

e =
( ∑

g∈Z(G)

αgg
)
E and e ∈ Q(Z(G)). Since Z(G) = {1, s}, the only possibilities for e

are

0, 1,
1 + s

2
and

1− s
2

.

Therefore, e = E and the component QG(1− Ĝ′) = QG(E) is simple, as we wanted to
prove.

4. Main Results

Now that we have introduced the terminology, fix the notation and display the auxiliary
results, we are able to present the main results. We begin describing U(ZG1), the group
of units of ZG1.



C. Mello / Eur. J. Pure Appl. Math, 10 (5) (2017), 955-966 959

4.1. The Group of Units of Group Ring ZG1

Proposition 5. The group ring QG1 admits the following decomposition:

QG1 = QG1

(1 + s

2

)
⊕QG1

(1− s
2

)
,

where QG1

(1 + s

2

)
∼= Q16 and QG1

(1− s
2

)
∼= M4(Q).

Proof. It follows from Proposition 1 and Proposition 4.

For the next results, ⊗ denotes the Kronecker product and we use the representation
of D4 on M2(Q) previously fixed and we obtain a representation of G1 on M4(Q):

• b 7→ e11 + e22 − e33 − e44 =

[
1 0
0 1

]
⊗
[

1 0
0 −1

]
,

• v 7→ −e13 − e24 + e31 + e42 =

[
1 0
0 1

]
⊗
[

0 −1
1 0

]
,

• b1 7→ e11 − e22 + e33 − e44 =

[
1 0
0 −1

]
⊗
[

1 0
0 1

]
,

• v1 7→ −e12 + e21 − e34 + e43 =

[
0 −1
1 0

]
⊗
[

1 0
0 1

]
.

Thus, the following proposition gives us an elementary Q-basis matrix of QG1

(1− s
2

)
∼=

M4(Q).

Proposition 6. Writing E =
1− s

2
and using the above representation, an elementary

Q-basis matrix of QG1

(1− s
2

)
∼= M4(Q) is:

e11 =
(1 + b

2

)(1 + b1
2

)
E, e12 =

(1 + b

2

)(v1b1 − v1
2

)
E,

e13 =
(vb− v

2

)(1 + b1
2

)
E, e14 =

(vb− v
2

)(v1b1 − v1
2

)
E,

e21 =
(1 + b

2

)(v1 + v1b1
2

)
E, e22 =

(1 + b

2

)(1− b1
2

)
E,

e23 =
(vb− v

2

)(v1 + v1b1
2

)
E, e24 =

(vb− v
2

)(1− b1
2

)
E,

e31 =
(v + vb

2

)(1 + b1
2

)
E, e32 =

(v + vb

2

)(v1b1 − v1
2

)
E,

e33 =
(1− b

2

)(1 + b1
2

)
E, e34 =

(1− b
2

)(v1b1 − v1
2

)
E,

e41 =
(v + vb

2

)(v1 + v1b1
2

)
E, e42 =

(v + vb

2

)(1− b1
2

)
E,

e43 =
(1− b

2

)(v1 + v1b1
2

)
E, e44 =

(1− b
2

)(1− b1
2

)
E.



C. Mello / Eur. J. Pure Appl. Math, 10 (5) (2017), 955-966 960

For the next result, we will need the following:

Definition 2. Let A =
[
2[Xij ] + I4

]
be a 4-by-4 matrix with Xij ∈ Z, for all i, j,

1 ≤ i, j ≤ 4. We define 4 distinct blocks of A:

B1 = {X11, X22, X33, X44},
B2 = {X12, X21, X34, X43},
B3 = {X13, X24, X31, X42},
B4 = {X14, X23, X32, X41}.

In particular, Bk =
{
Xiji | 1 ≤ i ≤ 4 and ji 6= ji′ , if i 6= i′

}
, 1 ≤ k ≤ 4.

Finally we are in a position to give a description of U(ZG1) in a similar vein as the
description given for U(ZD4) in Theorem 1 :

Theorem 2. Let G1 = D×D1
/
{1, ss1} be the extra-special 2-group of order 32, the central

product of two copies of D4 =
〈
b, v| b2 = v4 = 1 and bvbv = 1

〉
. Then U(ZG1) = ±G1V1

and V1 is isomorphic to the group [
2 [Xij ] + I4

]
det=1

of 4-by-4 matrices, where, for each k, 1 ≤ k ≤ 4, the integers Xiji ∈ Bk have the same
parity and 4 divides the sum X1j1 +X2j2 +X3j3 +X4j4.

Proof. By Proposition 3, U(ZG1) = ±G1V1, where V1 ∼= U2/G
′
1 and

U2 = U(ZG1) ∩
(
QG1

(1− s
2

)
+
(1 + s

2

))
=

= {u = 1 + α(1− s)|u ∈ U(ZG1), α ∈ ZG1}.
Thus, to describe U(ZG1) we need a complete description of V1 ∼= U2/G

′
1 of the U2,

that is, we need to describe completly the subgroup U2 of the U(ZG1).

Let u ∈ U2. Then, by Proposition 3 and writing E =
1− s

2
,

u = 1 + α(1 − s) = 1 + 2(α1 + α2 b + α3 v + α4 b1 + α5 v1 + α6 u + α7 u1 + α8 bb1 +
α9 bu1 + α10 bv1 + +α11 ub1 + α12 uu1 + α13 uv1 + α14 vb1 + α15 vu1 + α16 vv1)E.

The Proposition 5 gives QG1(E) ∼= M4(Q) and using the elementary basis matrix of
QG1(E) ∼= M4(Q) and the given representation of G1 in M4(Q), we obtain

u =
(1 + s

2

)
+ (e11 + e22 + e33 + e44) + 2[α1(e11 + e22 + e33 + e44) +α2(e11 + e22− e33−

e44) +α3(−e13− e24 + e31 + e42) +α4(e11− e22 + e33− e44) +α5(−e12 + e21− e34 + e43) +
α6(e13 + e24 + e31 + e42) + α7(e12 + e21 + e34 + e43) + α8(e11 − e22 − e33 + e44) + α9(e12 +
e21− e34− e43) +α10(−e12 + e21 + e34− e43) +α11(e13− e24 + e31− e42) +α12(e14 + e23 +
e32 + e41) + α13(−e14 + e23 − e32 + e41) + α14(−e13 + e24 + e31 − e42) + α15(−e14 − e23 +
e32 + e41) + α16(e14 − e23 − e32 + e41)].

Hence



C. Mello / Eur. J. Pure Appl. Math, 10 (5) (2017), 955-966 961

u =
(1 + s

2

)
+[2(α1+α2+α4+α8)+1]e11+2(−α5+α7+α9−α10)e12+2(−α3+α6+α11−

α14)e13+2(α12−α13−α15+α16)e14+2(α5+α7+α9+α10)e21+[2(α1+α2−α4−α8)+1]e22+
2(α12+α13−α15−α16)e23+2(−α3+α6−α11+α14)e24+2(α3+α6+α11+α14)e31+2(α12−
α13+α15−α16)e32+[2(α1−α2+α4−α8)+1]e33+2(−α5+α7−α9+α10)e34+2(α12+α13+
α15+α16)e41+2(α3+α6−α11−α14)e42+2(α5+α7−α9−α10)e43+[2(α1−α2−α4+α8)+1]e44.

Therefore u can be written as an integral invertible matrix U = [uij ] , 1 ≤ i, j ≤ 4,
where

u11 = 2(α1 + α2 + α4 + α8) + 1, u12 = 2(−α5 + α7 + α9 − α10),
u13 = 2(−α3 + α6 + α11 − α14), u14 = 2(α12 − α13 − α15 + α16),
u21 = 2(α5 + α7 + α9 + α10), u22 = 2(α1 + α2 − α4 − α8) + 1,
u23 = 2(α12 + α13 − α15 − α16), u24 = 2(−α3 + α6 − α11 + α14),
u31 = 2(α3 + α6 + α11 + α14), u32 = 2(α12 − α13 + α15 − α16),
u33 = 2(α1 − α2 + α4 − α8) + 1, u34 = 2(−α5 + α7 − α9 + α10),
u41 = 2(α12 + α13 + α15 + α16), u42 = 2(α3 + α6 − α11 − α14),
u43 = 2(α5 + α7 − α9 − α10), u44 = 2(α1 − α2 − α4 + α8) + 1.

Thus we produce the monomorphism

ϕ : U2 →


2Z + 1 2Z 2Z 2Z

2Z 2Z + 1 2Z 2Z
2Z 2Z 2Z + 1 2Z
2Z 2Z 2Z 2Z + 1


det=±1

defined by ϕ(u) =
[
2(Xij) + I4

]
, where

X11 = α1 + α2 + α4 + α8, X12 = −α5 + α7 + α9 − α10,
X13 = −α3 + α6 + α11 − α14, X14 = α12 − α13 − α15 + α16,
X21 = α5 + α7 + α9 + α10, X22 = α1 + α2 − α4 − α8,
X23 = α12 + α13 − α15 − α16, X24 = −α3 + α6 − α11 + α14,
X31 = α3 + α6 + α11 + α14, X32 = α12 − α13 + α15 − α16,
X33 = α1 − α2 + α4 − α8, X34 = −α5 + α7 − α9 + α10,
X41 = α12 + α13 + α15 + α16, X42 = α3 + α6 − α11 − α14,
X43 = α5 + α7 − α9 − α10, X44 = α1 − α2 − α4 + α8.

Let A = [2(Xij) + I4], with Xij ∈ Z, for all i, j, 1 ≤ i, j ≤ 4. Then

detA = 1 + 2β1 + 4β2 + 8β3 + 16β4,

where βr ∈ Z, 1 ≤ r ≤ 4. In particular, β1 = X11 + X22 + X33 + X44. Furthermore,
A ∈ ϕ(U2) if and only if detA = 1 and, for each k, 1 ≤ k ≤ 4, the integers Xiji ∈ Bk have
the same parity and 4 divides the sum X1j1 + X2j2 + X3j3 + X4j4 . Indeed, A ∈ ϕ(U2) if
and only if detA = ±1 and, for each k, 1 ≤ k ≤ 4, with Xiji ∈ Bk, 4 divides the sum



C. Mello / Eur. J. Pure Appl. Math, 10 (5) (2017), 955-966 962

X1j1 + X2j2 + X3j3 + X4j4 , this is, 4|(X1j1 + X2j2 + X3j3 + X4j4), and 2|(X1j1 + X2j2),
2|(X1j1 +X3j3), 2|(X1j1 +X4j4), 2|(X2j2 +X3j3), 2|(X2j2 +X4j4) and 2|(X3j3 +X4j4). In
particular, as 4|(X11 +X22 +X33 +X44), it follows that detA = 1.

So

ϕ(U2) =
[
2(Xij) + I4

]
det=1

is a group of 4-by-4 matrices, where for each k, 1 ≤ k ≤ 4, the integers Xiji ∈ Bk have
the same parity and 4 divides the sum X1j1 +X2j2 +X3j3 +X4j4 . Since

ϕ(s) = ϕ(1 + s(1− s)) = −I4,
it follows that the mapping ϕ induces an isomorphism from V1 onto the group[

2(Xij) + I4
]
det=1

of 4-by-4 matrices, where for each k, 1 ≤ k ≤ 4, the integers Xiji ∈ Bk have the same
parity and 4 divides the sum X1j1 +X2j2 +X3j3 +X4j4 .

4.2. The Group of Units of Group Ring ZG2

Proposition 7. The group ring QG2 admits the following decomposition:

QG2 = QG2

(1 + s

2

)
⊕QG2

(1− s
2

)
,

where QG2

(1 + s

2

)
∼= Q64 and QG2

(1− s
2

)
∼= M8(Q).

Proof. It follows from Proposition 4 and Proposition 5.

For the next results, we use the representation of D4 on M2(Q) previously fixed and
we obtain a representation of G2 on M8(Q):

• b 7→ e11 + e22 + e33 + e44− e55− e66− e77− e88 =

[
1 0
0 1

]
⊗
[

1 0
0 1

]
⊗
[

1 0
0 −1

]
,

• v 7→ −e15−e26−e37−e48+e51+e62+e73+e84 =

[
1 0
0 1

]
⊗
[

1 0
0 1

]
⊗
[

0 −1
1 0

]
,

• b1 7→ e11+e22−e33−e44+e55+e66−e77−e88 =

[
1 0
0 1

]
⊗
[

1 0
0 −1

]
⊗
[

1 0
0 1

]
,

• v1 7→ −e13−e24+e31+e42−e57−e68+e75+e86 =

[
1 0
0 1

]
⊗
[

0 −1
1 0

]
⊗
[

1 0
0 1

]
,

• b2 7→ e11−e22+e33−e44+e55−e66+e77−e88 =

[
1 0
0 −1

]
⊗
[

1 0
0 1

]
⊗
[

1 0
0 1

]
,

• v2 7→ −e12+e21−e34+e43−e56+e65−e78+e87 =

[
0 −1
1 0

]
⊗
[

1 0
0 1

]
⊗
[

1 0
0 1

]
.



C. Mello / Eur. J. Pure Appl. Math, 10 (5) (2017), 955-966 963

Just as extend the Proposition 2 can also extend the Proposition 6 and get an elemen-

tary Q-basis matrix of QG2

(1− s
2

)
∼= M8(Q), that will be used in next result. Even to

the next result, we will need the following definition:

Definition 3. Let A =
[
2(Xij) + I8

]
be a 8-by-8 matrix with Xij ∈ Z, for all i, j,

1 ≤ i, j ≤ 8. We define 8 distinct blocks of A:

B1 = {X11, X22, X33, X44, X55, X66, X77, X88},
B2 = {X12, X21, X34, X43, X56, X65, X78, X87},
B3 = {X13, X24, X31, X42, X57, X68, X75, X86},
B4 = {X14, X23, X32, X41, X58, X67, X76, X85},
B5 = {X15, X26, X37, X48, X51, X62, X73, X84},
B6 = {X16, X25, X38, X47, X52, X61, X74, X83},
B7 = {X17, X28, X35, X46, X53, X64, X71, X82},
B8 = {X18, X27, X36, X45, X54, X63, X72, X81}.

In particular, Bk =
{
Xiji | 1 ≤ i ≤ 8 and ji 6= ji′ , if i 6= i′

}
, 1 ≤ k ≤ 8.

Finally, with the same idea that we use to extend the Theorem 1, extend the Theorem
2 and describe completly U(ZG2):

Theorem 3. Let G2 = D × D1 × D2
/
{1, ss1, ss2, s1s2} be the extra-special 2-group of

order 128, the central product of three copies of

D4 =
〈
b, v| b2 = v4 = 1 and bvbv = 1

〉
.

Then U(ZG2) = ±G2V2 and V2 is isomorphic to the group[
2(Xij) + I8

]
det=1

of 8-by-8 matrices, where, for each k, 1 ≤ k ≤ 8, the integers Xiji ∈ Bk satisfy the
following conditions:

(i) All integers Xiji, 1 ≤ i ≤ 8, have the same parity;

(ii) 4 divides the sums X1j1 +X2j2 +X3j3 +X4j4 and X5j5 +X6j6 +X7j7 +X8j8;

(iii) If X1j1 and X(1+k)j(1+k)
, 1 ≤ k ≤ 3, are congruentes module 4, then X5j5 e X(5+k)j(5+k)

also are;

(iv) 8 divides the sum X1j1 +X2j2 +X3j3 +X4j4 +X5j5 +X6j6 +X7j7 +X8j8.

Proof. By Proposition 3, U(ZG2) = ±G2V2, where V2 ∼= U2/G
′
2 and

U2 = U(ZG2) ∩
(
QG2

(1− s
2

)
+
(1 + s

2

))
=

= {u = 1 + α(1− s)|u ∈ U(ZG2), α ∈ ZG2}.
Thus, to describe U(ZG2) we need a complet description of V2 ∼= U2/G

′
2 of the U2,

that is, we need to describe completly the subgroup U2 of the U(ZG2).



C. Mello / Eur. J. Pure Appl. Math, 10 (5) (2017), 955-966 964

Let u ∈ U2. Then, by Proposition 3 and writing E =
1− s

2
,

u = 1 + α(1 − s) = 1 + 2(α1 + α2 b + α3 v + α4 b1 + α5 v1 + α6 b2 + α7 v2 + α8 u +
α9 u1 + +α10 u2 + α11 bb1 + α12 bu1 + α13 bv1 + α14 ub1 + α15 uu1 + α16 uv1 + α17 vb1 +
α18 vu1 + α19 vv1 + +α20 bb2 + α21 bu2 + α22 bv2 + α23 ub2 + α24 uu2 + α25 uv2 + α26 vb2 +
α27 vu2 + α28 vv2 + +α29 b1b2 + α30 b1u2 + α31 b1v2 + α32 u1b2 + α33 u1u2 + α34 u1v2 +
α35 v1b2 +α36 v1u2 ++α37 v1v2 +α38 bb1b2 +α39 bb1u2 +α40 bb1v2 +α41 bu1b2 +α42 bu1u2 +
α43 bu1v2++α44 bv1b2+α45 bv1u2+α46 bv1v2+α47 ub1b2+α48 ub1u2+α49 ub1v2+α50 uu1b2+
+α51 uu1u2 + α52 uu1v2 + α53 uv1b2 + α54 uv1u2 + α55 uv1v2 + α56 vb1b2 + α57 vb1u2 +
+α58 vb1v2 + α59 vu1b2 + α60 vu1u2 + α61 vu1v2 + α62 vv1b2 + α63 vv1u2 + α64 vv1v2)E.

The Proposition 7 gives QG2(E) ∼= M8(Q) and using the elementary matrix basis of
QG1(E) ∼= M8(Q) and the give representation of G2 in M8(Q), u can be written as an
integral invertible matrix [uij ] , 1 ≤ i, j ≤ 8. Thus we produce the monomorphism defined

by ϕ(u) =
[
2(Xij) + I8

]
.

Let A = [2(Xij) + I8], with Xij ∈ Z, for all i, j, 1 ≤ i, j ≤ 8. Then

detA = 1 + 2β1 + 4β2 + 8β3 + 16β4 + 32β5 + 64β6 + 128β7 + 256β8,

where βr ∈ Z, 1 ≤ r ≤ 8. In particular,

β1 = X11 +X22 +X33 +X44 +X55 +X66 +X77 +X88.

Furthermore, A ∈ ϕ(U2) if and only if detA = 1 and, for each k, 1 ≤ k ≤ 8, the
integers Xiji ∈ Bk satisfy the following conditions:

(i) All integers Xiji , 1 ≤ i ≤ 8, have the same parity;

(ii) 4 divides the sums X1j1 +X2j2 +X3j3 +X4j4 and X5j5 +X6j6 +X7j7 +X8j8 ;

(iii) IfX1j1 andX(1+k)j(1+k)
, 1 ≤ k ≤ 3, are congruentes module 4, thenX5j5 eX(5+k)j(5+k)

also are;

(iv) 8 divides the sum X1j1 +X2j2 +X3j3 +X4j4 +X5j5 +X6j6 +X7j7 +X8j8 .

Indeed, A ∈ ϕ(U2) if and only if detA = 1 and, for each k, 1 ≤ k ≤ 8, the integers
Xiji ∈ Bk satisfy the following conditions:

• 2 divides the sum Xiji +Xi′ji′
, for all 1 ≤ i, i′ ≤ 8;

• 4 divides the following sums:

· X1j1 +X2j2 +Xiji +X(i+1)j(i+1)
, with i ∈ {3, 5, 7};

· X1j1 +X3j3 +Xiji +X(i+2)j(i+2)
, with i ∈ {5, 6};

· X1j1 +X4j4 +X5j5 +X8j8 ;

· X1j1 +X4j4 +X6j6 +X7j7 ;

· X2j2 +X3j3 +X5j5 +X8j8 ;



REFERENCES 965

· X2j2 +X3j3 +X6j6 +X7j7 ;

· X2j2 +X4j4 +Xiji +X(i+2)j(i+2)
, with i ∈ {5, 6};

· X3j3 +X4j4 +Xiji +X(i+1)j(i+1)
, with i ∈ {5, 7};

· X5j5 +X6j6 +X7j7 +X8j8 .

• 8 divides the sum X1j1 +X2j2 +X3j3 +X4j4 +X5j5 +X6j6 +X7j7 +X8j8 .

In particular, as 8 divides X11+X22+X33+X44+X55+X66+X77+X88, we have that
detA = 1. Furthermore, ϕ(s) = ϕ(1 + s(1− s)) = −I8. So, it follows that the mapping ϕ
induces the isomorphism wanted.

The reader may notice that the idea used in this paper can be extended to describe the
group of units of any integral group ring of a finite extra-special 2-group of order higher
than 128, that is a central product of copies of D4.

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