Unit groups of integral finite group rings with no noncyclic abelian
  finite subgroups
UNIT GROUPS OF INTEGRAL FINITE GROUP RINGS WITH
NO NONCYCLIC ABELIAN FINITE SUBGROUPS
MARTIN HERTWECK
Abstract. It is shown that in the units of augmentation one of an integral
group ring ZG of a finite group G, a noncyclic subgroup of order p2, for some
odd prime p, exists only if such a subgroup exists in G. The corresponding
statement for p = 2 holds by the Brauer–Suzuki theorem, as recently observed
by W. Kimmerle.
1. Introduction
Is a finite subgroup H of units in the integral group ring ZG of a finite group
G necessarily isomorphic to a subgroup of G? Of course, torsion coming from the
coefficient ring should be excluded, that is, only finite subgroups H in V(ZG), the
group of units of augmentation one in ZG, will be considered. The question was
raised by Higman in his thesis (1940), where he gave an affirmative answer when G
is metabelian nilpotent or the affine group over a prime field; cf. Sandling (1981).
In the survey of Sandling (1984) it is included as Problem 5.4, and noted that an
affirmative answer for metabelian G was finally given by Roggenkamp (1981); but
see also Cliff, Sehgal and Weiss (1981), and Marciniak and Sehgal (2003) for a more
recent result, giving a generalization based on a theorem of Weiss (1988). These
results are really about certain ‘large’ torsion-free normal subgroups of V(ZG). For
a more complete discussion, see Chapter 4 in Sehgal’s book (1993).
As a sort of converse, one may fix a finite group H and look for groups G for
which H embeds into V(ZG), again hoping for the best, but little is known in this
respect. What is known is that if a cyclic group H of prime power order embeds
into some unit group V(ZG), then H also embeds into G (due to an observation
of Cohn and Livingstone (1965); see also Zassenhaus (1974)), and only recently in
Hertweck (2007b) it was shown that the restriction on the order can be removed
if in addition G is assumed to be solvable. In this spirit, Marciniak, at a satellite
conference of the ICM 2006, asked whether a group G necessarily has a subgroup
isomorphic to Klein’s four group provided this is the case for V(ZG). Kimmerle
immediately observed that this is implied by the Brauer–Suzuki theorem (rendered
in Kimmerle (2006)), see Section 2. Our complementary result is as follows.
Theorem A. Let G be a finite group. Suppose that V(ZG) has a noncyclic abelian
subgroup of order p2, for some odd prime p. Then the same is true for G (i.e.,
Sylow p-subgroups of G are not cyclic).
Date: October 30, 2018.
2000 Mathematics Subject Classification. Primary 16S34, 16U60; Secondary 20C05.
Key words and phrases. integral group ring, torsion unit, partial augmentation.
http://arxiv.org/abs/0704.0412v1
2 MARTIN HERTWECK
It is easy to verify that a finite p-group with no noncyclic abelian subgroup is
either cyclic or a (generalized) quaternion group, see Theorem 4.10 in Gorenstein
(1968). It comes to mind that the theory of cyclic blocks might be used in the
proof, but it is pretty simple and makes only use of a fact about vanishing of
partial augmentations of torsion units, established in Hertweck (2006, 2007a).
We remark that both results (whether p is even or odd) for a solvable group G
are covered by Theorem 5.1 in Dokuchaev and Juriaans (1996).
Note that a group G whose Sylow 2-subgroups are cyclic has a normal 2-comple-
ment, by Burnside’s well known criterion, see Theorem 4.3 in Gorenstein (1968).
We obtain the following corollary.
Corollary 1. Let G be a finite group having cyclic Sylow p-subgroups for some
prime p. Then any finite p-subgroup of V(ZG) is isomorphic to a subgroup of G.
Finally, we remark that, as with other results in this field, the theorem can be
formulated for more general coefficient rings than Z, notably for the semilocalization
of Z at the prime divisors of the order of G. Unfortunately, it is definitely wrong
for p-adic coefficient rings.
2. Kimmerle’s observation
Coming back to the initial question, we mention that in the hope for further
positive results, it is natural to impose restrictions on the prime divisors of the
finite subgroup H , i.e., to consider only π-groups H for some set π of primes (a
singleton {p}, to begin with), as has been done before in work on the stronger
Zassenhaus conjecture (ZC3), cf. Dokuchaev and Juriaans (1996). It is well known
that then, one can assume that Oπ′(G), the largest normal π
′-subgroup of G, is
trivial, for H has an isomorphic image under the natural map ZG → ZG/Oπ′(G),
see the remark after Theorem 2.2 in Dokuchaev and Juriaans (1996).
This derives from the vanishing of certain partial augmentations of the elements
of H . Recall that for a group ring element u =
g∈G agg (all ag in Z), its partial
augmentation with respect to an element x of G, or rather its conjugacy class xG
in G, is the sum
g∈xG ag; we will denote it by εx(u). The result of Cohn and
Livingstone mentioned in the introduction really says that if an element h of H is
of prime power order, then there exists an element x in G of the same order such
that εx(h) 6= 0. Note that εz(u) = az for an element z in the center of G. An
old yet fundamental result from Berman (1955) and Higman (1940) asserts that if
εz(h) 6= 0 for an element h in H and some z in the center of G, then h = z.
Coming to Marciniak’s question, suppose that G has no subgroups isomorphic to
Klein’s four group. For our purpose, we can assume that O2′(G) = 1 and that Sylow
2-subgroups of G are not cyclic. Thus Sylow 2-subgroups of G are (generalized)
quaternion, and by the Brauer–Suzuki theorem, from Brauer and Suzuki (1959), G
contains a unique involution z. For an involution u in V(ZG), the Cohn–Livingstone
result gives εz(u) 6= 0, and therefore u = z by the Berman–Higman result, answering
Marciniak’s question in the affirmative.
Theorem B (Kimmerle). Let G be a finite group. Suppose that V(ZG) has a
subgroup isomorphic to Klein’s four group. Then the same is true for G.
We do not know of a proof avoiding the use of the Brauer–Suzuki theorem.
Suppose that Sylow 2-subgroups of G are quaternion groups. Then the theorem
implies that finite 2-subgroups of V(ZG) are cyclic or quaternion groups. Taking
UNIT GROUPS WITH NO NONCYCLIC ABELIAN FINITE SUBGROUPS 3
into account the structure of the quaternion groups, and the Cohn–Livingstone
result, one obtains the following corollary.
Corollary 2. Let G be a finite group whose Sylow 2-subgroups are quaternion
groups (ordinary or generalized). Then any finite 2-subgroup of V(ZG) is isomor-
phic to a subgroup of G.
3. Proof of Theorem A
The partial augmentations of a torsion unit in V(ZG) encode its character values
in a way establishing a connection to group elements which respects a divisibility
relation between orders. We will make use of a lemma which is an easy consequence
of this fact.
Lemma 3. Let u be a torsion unit in V(ZG) of, say, order n. Let s be a natural
integer coprime to n, so that st ≡ 1 mod n for another natural integer t. Then for
all x in G whose order divide n, we have εx(u
s) = εxt(u).
Proof. Let ζ be a primitive n-th complex root of unity, and let σ be the Galois auto-
morphism ofQ(ζ) sending ζ to ζs. Let x1, . . . , xk be representatives of the conjugacy
classes of G whose elements have order dividing n. Note that then xt1, . . . , x
k is an-
other system of representatives. By Theorem 2.3 in Hertweck (2007a), εx(u) 6= 0 is
possible only for elements x whose order divide n. Thus for any ordinary irreducible
character χ of G, we have
εxi(u
s)χ(xi) = χ(u
s) = χ(u)σ =
εxi(u)χ(xi)
εxi(u)χ(x
i ) =
(u)χ(xi).
Since the character table of G, stripped off from any additional information, is an
invertible matrix, it follows that εxi(u
s) = εxt
(u) for all indices i, which proves the
lemma. �
Corollary 4. Let u be a torsion unit in V(ZG) of, say, order n. Then for any x
in G whose order divides n,
s∈(Z/nZ)×
s∈(Z/nZ)×
εxs(u).
Corollary 5. Suppose that for a prime divisor p of the order of G, all elements
of order p in G are conjugate to a power of some fixed element x. Let u be a
torsion unit in V(ZG) of order p. Then
i=1 u
i and
i=1 x
i have the same
partial augmentations.
Proof. Let k be the number of conjugacy classes of elements of order p in G. By
Corollary 4 and Theorem 2.3 in Hertweck (2007a),
εxi(u) =
yG : y∈〈x〉
εy(u) =
Applying again Theorem 2.3 from Hertweck (2007a), the corollary follows. �
4 MARTIN HERTWECK
We will apply this by means of the following formula relating ranks of an idem-
potent to arithmetical properties of the group.
Corollary 6. Suppose that for a prime divisor p of the order of G, all elements
of order p in G are conjugate to a power of some fixed element x. Suppose further
that V(ZG) contains an elementary abelian subgroup U of order p2. Then for any
ordinary character χ of G,
(1) χ
χ(1) + (p+ 1)
χ(xi)
We now turn to the proof of Theorem A. Suppose that G has a cyclic Sylow
p-subgroup P (p = 2 is allowed). Let x be an element of order p in P , and
set N = NG(〈x〉). Suppose further that V(ZG) contains an elementary abelian
subgroup U of order p2. Let χ be the character of G which is induced from the
principal irreducible character of P . Then the rank in (1) is
(|G : P |+ |N : P |(p2 − 1)).
If χ is a character of G which is induced from a faithful irreducible character of P ,
the rank in (1) is
(|G : P | − |N : P |(p+ 1)).
The difference of these ranks is |N : P |(p2 + p)/p2, which is not an integer. This
contradiction proves the theorem.
In view of Corollaries 1 and 2, one may be tempted to investigate the analogous
problem for groups with dihedral Sylow 2-subgroups. These groups were classified
by Gorenstein and Walter, and listed, for example, on p. 462 in Gorenstein (1968).
To indicate what can be done by now, we end with an example.
Note that the order of a finite subgroup of V(ZG) divides the order of G, see
Lemma 37.3 in Sehgal (1993); a fact which, surprisingly enough from today’s point
of view, is in this generality not recorded in Higman’s thesis.
Example 7. For the alternating group A7, any finite 2-subgroup of V(ZA7) is
isomorphic to a subgroup of A7.
Proof. Sylow 2-subgroups of A7 are dihedral of order 8. Let x be an element of
order 4 in A7. Then x
G and (x2)G are the only conjugacy classes of elements of
order 4 and 2, respectively. There is an (irreducible) character χ of A7 of degree 6
which is afforded by a deleted permutation representation. We have χ(x) = 0 and
χ(x2) = 2.
Let U be a finite 2-subgroup of V(ZA7). If U is of order 2, then U is rationally
conjugate to a subgroup of A7 by Corollary 3.5 in Hertweck (2006). If U is of
order 4, the Luthar–Passi method as described in Hertweck (2007a) is not sufficient
to guarantee rational conjugacy to a subgroup of A7: for a unit u of order 4 in
V(ZA7) one cannot exclude the possibility of having (εx2(u), εx(u)) = (2,−1) when
χ(u) = 4. In this case, also χ(u−1) = 4. Anyway, U is isomorphic to a subgroup of
A7, and the same is true if U is a Klein’s four group.
Suppose that U is abelian of order 8. By the Cohn–Livingstone result, U is not
cyclic. Set e = 1
u∈U u. Since e is an idempotent, χ(u) is a rational integer. If
U is elementary abelian, then χ(e) = 1
(χ(1) + 7χ(x2)) = 20
, which is impossible.
UNIT GROUPS WITH NO NONCYCLIC ABELIAN FINITE SUBGROUPS 5
Thus U contains 3 elements of order 2 and 4 elements of order 4. Trying out all
possibilities shows that again χ(e) is not a rational integer.
It remains to consider the case when U is the quaternion group. Let u be an
element of order 4 in U . Since χ(u2) = χ(x2), the restriction of the character χ
to U is the sum of four linear characters and the one of degree two. But this is
impossible since χ is afforded by a rational representation, while the character of
degree two of the quaternion group comes from the block of the rational quaternion
algebra (whence the name of the group). �
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Universität Stuttgart, Fachbereich Mathematik, IGT, Pfaffenwaldring 57, 70550
Stuttgart, Germany
E-mail address: hertweck@mathematik.uni-stuttgart.de
arXiv:math.RA/0612429v2
arXiv:math.RT/0703541
	1. Introduction
	2. Kimmerle's observation
	3. Proof of Theorem A
	References
