Abnormal subgroups and Carter subgroups in some infinite groups
t. Some properties of abnormal subgroups in generalized soluble groups are considered. In particular, the transitivity of abnormality in metahypercentral groups is proven. Also it is proven that a subgroup H of a radical group G is abnormal in G if and only if every intermediate subgroup for H co...
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| Цитувати: | Abnormal subgroups and Carter subgroups in some infinite groups / L.A. Kurdachenko, I.Ya. Subbotin // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 69–83. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1566102019-06-19T01:29:31Z Abnormal subgroups and Carter subgroups in some infinite groups Kurdachenko, L.A. Subbotin, I.Ya. t. Some properties of abnormal subgroups in generalized soluble groups are considered. In particular, the transitivity of abnormality in metahypercentral groups is proven. Also it is proven that a subgroup H of a radical group G is abnormal in G if and only if every intermediate subgroup for H coincides with its normalizer in G. This result extends on radical groups the wellknown criterion of abnormality for finite soluble groups due to D. Taunt. For some infinite groups (not only periodic) the existence of Carter subgroups and their conjugation have been also obtained. 2005 Article Abnormal subgroups and Carter subgroups in some infinite groups / L.A. Kurdachenko, I.Ya. Subbotin // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 69–83. — Бібліогр.: 25 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20E34, 20F19, 20F22. http://dspace.nbuv.gov.ua/handle/123456789/156610 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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t. Some properties of abnormal subgroups in generalized soluble groups are considered. In particular, the transitivity
of abnormality in metahypercentral groups is proven. Also it is
proven that a subgroup H of a radical group G is abnormal in G
if and only if every intermediate subgroup for H coincides with its
normalizer in G. This result extends on radical groups the wellknown criterion of abnormality for finite soluble groups due to D.
Taunt. For some infinite groups (not only periodic) the existence
of Carter subgroups and their conjugation have been also obtained. |
| format |
Article |
| author |
Kurdachenko, L.A. Subbotin, I.Ya. |
| spellingShingle |
Kurdachenko, L.A. Subbotin, I.Ya. Abnormal subgroups and Carter subgroups in some infinite groups Algebra and Discrete Mathematics |
| author_facet |
Kurdachenko, L.A. Subbotin, I.Ya. |
| author_sort |
Kurdachenko, L.A. |
| title |
Abnormal subgroups and Carter subgroups in some infinite groups |
| title_short |
Abnormal subgroups and Carter subgroups in some infinite groups |
| title_full |
Abnormal subgroups and Carter subgroups in some infinite groups |
| title_fullStr |
Abnormal subgroups and Carter subgroups in some infinite groups |
| title_full_unstemmed |
Abnormal subgroups and Carter subgroups in some infinite groups |
| title_sort |
abnormal subgroups and carter subgroups in some infinite groups |
| publisher |
Інститут прикладної математики і механіки НАН України |
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2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/156610 |
| citation_txt |
Abnormal subgroups and Carter subgroups in some infinite groups / L.A. Kurdachenko, I.Ya. Subbotin // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 69–83. — Бібліогр.: 25 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT kurdachenkola abnormalsubgroupsandcartersubgroupsinsomeinfinitegroups AT subbotiniya abnormalsubgroupsandcartersubgroupsinsomeinfinitegroups |
| first_indexed |
2025-07-14T09:00:13Z |
| last_indexed |
2025-07-14T09:00:13Z |
| _version_ |
1837612258406957056 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2005). pp. 69 – 83
c© Journal “Algebra and Discrete Mathematics”
Abnormal subgroups and Carter subgroups in
some infinite groups
L. A. Kurdachenko and I. Ya. Subbotin
Dedicated to Yu.A. Drozd on the occasion of his 60th birthday
Abstract. Some properties of abnormal subgroups in gener-
alized soluble groups are considered. In particular, the transitivity
of abnormality in metahypercentral groups is proven. Also it is
proven that a subgroup H of a radical group G is abnormal in G
if and only if every intermediate subgroup for H coincides with its
normalizer in G. This result extends on radical groups the well-
known criterion of abnormality for finite soluble groups due to D.
Taunt. For some infinite groups (not only periodic) the existence
of Carter subgroups and their conjugation have been also obtained.
A subgroup H of a group G is abnormal in G if g ∈ 〈H, Hg〉 for each
element g ∈ G. Abnormal subgroups have appeared in the paper [HP] due
to P. Hall, while the term "abnormal subgroup" itself belongs to R. Carter
[CR]. Abnormal subgroups are antipodes to normal subgroups. In finite
(especially soluble) groups the properties of abnormal subgroups have
been studied in details. However we cannot say this about infinite groups.
It is even unknown what groups containe proper abnormal subgroups. In
this connection recall that a finite group is nilpotent if and only if it does
not include proper abnormal subgroups. If G is a locally nilpotent group,
then G has no proper abnormal subgroups [KUS]. However, in general, the
converse assersion is not proven. In [KOS, KS 2] some classes of infinite
groups, in which the absence of proper abnormal subgroups implies locally
nilpotency, have been considered.
As normality, abnormality is not a transitive relation (see, for example
group S4). The groups (finite and infinite) with transitivity of normality
2000 Mathematics Subject Classification: 20E34, 20F19, 20F22.
Key words and phrases: abnormal subgroups, pronormal subgroups, Carter
subgroups.
70 Abnormal subgroups and Carter subgroups...
are studied well enough (the most complete results one can find in [RD]).
The situation for the groups with transitivity of abnormality is different.
From the results of § 13 of Chapter VI of book [HB] it follows that in
every finite metanilpotent group abnormality is transitive. For infinite
groups this question has been considered in [KS 1, KS 2]. In the current
article the following most general result has been obtained.
1.2. Theorem. Let G be a group and suppose that A is a normal
subgroup of G such that G/A has no proper abnormal subgroups. If A
satisfies the normalizer condition, then in G abnormality is transitive.
It is interesting to mention that this theorem and results of [RD]
implies that soluble groups with transitivity of normality is a proper
subclass of soluble groups in which abnormality is transitive.
In finite soluble groups abnormality is tightly connected to selfnor-
malizing. For example,D.Taunt has proved that a subgroup H of a finite
soluble group G is abnormal if and only if every intermediate subgroup
for H coincides with its normalizer in G (see, for example, [RD 6, 9.2.11]).
Remind that a subgroup S is said to be an intermediate subgroup for H
if H ≤ S. The following theorem extends this result on radical groups.
1.6. Theorem. Let G be a radical group and let H be a subgroup of
G. Then H is abnormal in G if and only if every intermediate subgroup
for H coincides with its normalizer in G.
In finite groups there are many important families of subgroups hav-
ing crucial influence on the group structure (for example Sylow and Hall
subgroups, system normalizers, subgroups defined by formations, and
so on). Many of them are very specific for the finite groups. With
Carter subgroups the situation is different. In their original definition we
can find no specifications related to finite groups. These subgroups are
very tightly connected to abnormality. Indeed, Carter subgroups of finite
groups can be defined as abnormal nilpotent subgroups. The following
question naturally arises: which infinite groups posses Carter subgroups?
S.E. Stonehewer in his papers [SE 1, SE 2] proved that periodic locally
soluble groups having a locally nilpotent radical of finite index and locally
soluble FC – groups are some examples of such infinite groups. In the
current article we consider the following class of infinite groups.
Let X is a class of groups. A group G is said to be an artinian–by
–X– group if G has a normal subgroup H such that G/H belongs to the
class X and H satisfies Min–G.
L. A. Kurdachenko, I. Ya. Subbotin 71
This kind of groups has been introduced by D.J.S. Robinson [RD 3
– RD 5, RD 7, RD 8] and D.I. Zaitsev [ZD 1 – ZD 5] in their series
of papers dedicated to the existence of complements to the X – residual
(when it is abelian) for some natural classes X (such as hypercentral
groups, locally nilpotent groups, hypercyclic groups, locally supersoluble
groups, hyperfinite groups). For the classes X considered in our paper
this definition implies that if G is an artinian – by –X– group and R = GX
is its X– residual, then G/R ∈ X and R satisfies Min–G.
We will deal with artinian – by – hypercentral groups whose locally
nilpotent residual is nilpotent. It is a natural first step. Since these
groups are generalizations of finite metanilpotent groups, we will use
for the definition some characterizations of Carter subgroups, which are
valid for finite metanilpotent groups. In particular, in finite metanilpo-
tent groups Carter subgroups coincide with system normalizers (see, for
example, [RD 6, 9.5.10]). By a result due to P. Hall (see, for example,
[RD 6, 9.2.15]) the system normalizers of a finite soluble group are pre-
cisely its minimal subabnormal subgroups. As we have noted above, in
a finite metanilpotent group every subabnormal subgroup is abnormal.
Consequently, we can define a Carter subgroup in a metanilpotent group
as a minimal abnormal subgroup.
2.1. Theorem. Let G be an artinian – by – hypercentral group
and suppose that its locally nilpotent residual K is nilpotent. Then G
includes a minimal abnormal subgroup L. Moreover, L is a maximal
hypercentral subgroup and it includes the upper hypercenter of G. In
particular, G = KL. If H is another minimal abnormal subgroup, then
H is conjugate to L.
2.2. Corollary. Let G be an artinian – by – hypercentral group
and suppose that its locally nilpotent residual K is nilpotent. Then G
includes a hypercentral abnormal subgroup L. Moreover, L is maximal
hypercentral subgroup and it includes the upper hypercenter of G. In
particular, G = KL. If H is another hypercentral abnormal subgroup,
then H is conjugate to L.
Let G be an artinian – by – hypercentral group with a nilpotent
hypercentral residual.
A subgroup L is called a Carter subgroup of a group G if H is a
hypercentral abnormal subgroup of G (or, equivalently, H is a minimal
abnormal subgroup of G).
72 Abnormal subgroups and Carter subgroups...
A Carter subgroup in finite soluble group can be defined as a covering
subgroup for the formation of nilpotent groups. As we will see, this
characterization can be extended on the groups under consideration.
Recall that a subgroup H of a group G is said to be a LN -covering
subgroup if H is locally nilpotent and if S = SLNH for every subgroup
S, which includes H. Here SLN is a locally nilpotent residual of subgroup
S.
2.3. Theorem. Let G be an artinian – by – hypercentral group and
suppose that its locally nilpotent residual K is nilpotent. If L is a Carter
subgroup of G, then L is a LN – covering subgroup of G. Conversely,
if H is a LN – covering subgroup of G, then H is a Carter subgroup of
G.
In a finite soluble group N– covering subgroups are exactly N – pro-
jectors. Therefore a Carter subgroup of a finite soluble group can be
defined as an N – projector. As we will see, this characterization also can
be extended on the considered in this article groups.
Recall that a subgroup L of a group G is said to be a locally nilpotent
projector, if LH/H is a maximal locally nilpotent subgroup of G/H for
each normal subgroup H of a group G.
2.4. Theorem. Let G be an artinian – by – hypercentral group and
suppose that its locally nilpotent residual K is nilpotent. If L is a Carter
subgroup of G, then L is a locally nilpotent projector of G. Conversely,
if D is a locally nilpotent projector of G, then H is a Carter subgroup
of G.
1. Abnormal subgroups in some infinite groups
We will consider some classes of groups, in which abnormality is transi-
tive.
Lemma 1.1. Let G be a group and B be an abnormal subgroup of G.
Suppose that B = HA where A is a normal subgroup of a group G. If H
is an abnormal subgroup of B, then H is an abnormal subgroup of G.
This assertion belongs to P. Hall and one can find it, for example, in
the book [RD 6, 9.2.12] where it is proven for finite groups G. However
this proof does not use the finiteness of G and the result is valid for
infinite groups also.
L. A. Kurdachenko, I. Ya. Subbotin 73
Theorem 1.2. Let G be a group and suppose that A is a normal subgroup
of G such that G/A has no proper abnormal subgroups. If A satisfies the
normalizer condition, then in G abnormality is transitive
Proof. Let B be an abnormal subgroup of G and let H be an abnormal
subgroup of B. If B = G, then all is proved. Suppose that B 6= G.
By elementary properties of abnormal subgroups AB is abnormal in G.
Since G/A has no proper abnormal subgroup, AB = G. Proceeding by
induction, we will construct a strictly ascending chain
B = B0 < B1 < ...Bα < Bα+1 < ...
of subgroups such that H is abnormal in every subgroup Bα. Put K =
A ∩ B. Then K is a normal subgroup of B, so that KH is abnormal in
B and KH/K is abnormal in B/K. Since G/A = AB/A ∼= B/(A∩B) =
B/K, we obtain that B/K has no proper abnormal subgroups. It follows
that B/K = HK/K, that is B = HK. Since A satisfies the normalizer
condition, there exists a subgroup K1 such that K < K1 and K is a
normal subgroup of K1. Put B1 = 〈B, K1〉 . Then B 6= B1. Since K is
normal in both B and K1, K is normal in B1. By Lemma 1.1 the equation
B = HK implies that H is abnormal in B1. Suppose that we have already
constructed the subgroups Bβ for all β < a. Let first α − 1 exists and
Bα−1 6= G. Since B ≤ Bα−1, the latter is abnormal in G. By elementary
properties of abnormal subgroups ABα−1 is abnormal in G. Since G/A
has no proper abnormal subgroups, ABα−1 = G. Put Kα−1 = A∩Bα−1.
Then Kα−1 is a normal subgroup of Bα−1, so that Kα−1H is abnor-
mal in Bα−1 and Kα−1H/Kα−1 is abnormal in Bα−1/Kα−1. However,
since G/A = ABα−1/A ∼= Bα−1/(A ∩ Bα−1) = Bα−1/Kα−1,we obtain
that Bα−1/Kα−1 has no proper abnormal subgroups. It follows that
Bα−1/Kα−1 = Kα−1H/Kα−1, that is Bα−1 = Kα−1H. Since A satisfies
the normalizer condition, there exists a subgroup Kα such that Kα−1 <
Kα and Kα−1 is a normal subgroup of Kα. Put Bα = 〈Bα−1, Kα〉. Then
Bα−1 6= Bα. Since Kα−1 is normal in both Bα−1 and Kα , Kα−1 a is
normal in Bα. By Lemma 1.1 the equation Bα−1 = Kα−1H implies that
H is abnormal in Bα.
Suppose now that α is a limit ordinal. Then put Bα =
⋃
β<α
Bβ . Let
x be an arbitrary element of Bα. Then x ∈ Bν for some ν < α. We have
〈H, Hx〉 ≤ Bν . Since H is abnormal in Bν , x ∈ 〈H, Hx〉. This means
that H is an abnormal subgroup of Bα.
By our construction, Bα−1 6= Bα for each ordinal α. It follows that
there is an ordinal γ such that Bγ = G. Thus H is abnormal in Bγ =
G.
74 Abnormal subgroups and Carter subgroups...
Corollary 1.3. Let G be a group and suppose that A is a normal subgroup
of G such that G/A does not include proper abnormal subgroup. If A is
hypercentral, then in G abnormality is transitive.
Corollary 1.4. Let G be a group and suppose that A is a normal subgroup
of G such that G/A is locally nilpotent. If A is hypercentral, then in G
abnormality is transitive.
Indeed, a locally nilpotent group does not include proper abnormal
subgroups [KUS].
Lemma 1.5. Let G be a group, H be a subgroup of G and let D be an
H – invariant subgroup. Suppose that every intermediate subgroup for H
coincides with its normalizer in G. If D is a locally nilpotent subgroup,
then H is abnormal in HD.
Proof. Put L = HD. Choose an arbitrary element d ∈ D and consider
the subgroup K =
〈
H, Hd
〉
. Since H ≤ K, L = DK. It follows that
B = D∩K is a normal subgroup of K, in particular, B is H – invariant.
If d ∈ B, then d ∈ HB = K =
〈
H, Hd
〉
. Suppose that d /∈ B. In the
subgroup U = 〈d, B〉 choose a subgroup M, which is maximal with the
properties B ≤ M and d /∈ M . Clearly M is a maximal subgroup of U .
Since U is locally nilpotent, M is normal in U ( see, for example, [RD 1,
Theorem 5.38]). In particular, d−1Md = M . If h ∈ H then [h, d] ∈ K,
that is [h, d] ∈ D ∩ K. As above K = (D ∩ K)H = BH. We have now
hd = h[h, d] ∈ HB = K for each element h ∈ H. Let y ∈ M, h ∈ H and
consider the element d−1(h−1yh)d = d−1h−1yhd. Since dhd−1h−1 = b ∈
B, d−1h−1 = h−1d−1b. Similarly, hd = adh for some element a ∈ B. Now
we have d−1h−1yhd = h−1d−1(bya)dh. The inclusion B ≤ M implies that
bya ∈ M . Then d−1(bya)d ∈ M and hence h−1d−1(bya)dh. ∈ h−1Mh.
In other words, d−1(h−1Mh)d = h−1Mh. Let C =
⋂
h∈H
(h−1Mh.), then
by above d−1Cd = C. Since B ≤ M, B = h−1Bh. ≤ h−1Mh for each
h ∈ H, so that B ≤ C. Furthermore, d−1Hd ≤ K = HB ≤ HC.
It follows that d−1(HC)d ≤ HC. In other words, d ∈ NG(HC). Since
HC ∩ D = C(H ∩ D) ≤ CB = C, d /∈ HC. On the other hand, by the
conditions of our lemma HC is self – normalizing. This contradiction
shows that d ∈ B and hence d ∈ K =
〈
H, Hd
〉
, which means that H is
abnormal in HD.
Theorem 1.6. Let G be a radical group and let H be a subgroup of G.
Then H is abnormal in G if and only if every intermediate subgroup for
H coincides with its normalizer in G.
L. A. Kurdachenko, I. Ya. Subbotin 75
Proof. Let
〈1〉 = A0 ≤ A1 ≤ ...Aα ≤ Aα+1 ≤ ...Aγ = G
be a series of normal subgroups of G, whose factors are locally nilpotent.
If H is abnormal in G, then every intermediate subgroup for H is self
– normalizing (see, for example, [RD 6, p. 266]). Conversely, suppose
that every intermediate subgroup for H is self – normalizing and will
prove that H is abnormal in G. More precisely, we will prove that H
is abnormal in AαH for each α ≤ γ. By Lemma 1.5 H is abnormal in
HA1, so, for the case α = 1 all is proved. Suppose that we have already
proved that H is abnormal in AβH for all β < α. Choose an arbitrary
element a ∈ Aα and consider the subgroup K = 〈H, Ha〉. Suppose firstly
that α is a limit ordinal. Then there is an ordinal β < α such that
a ∈ Aβ . Since H is abnormal in AβH, a ∈ K = 〈H, Ha〉. It follows
that H is abnormal in AαH. Assume now that α is not limit. Put
U = Aα and V = Aα−1. Consider the factor – group UH/V . Let Z/V
be an intermediate subgroup for HV/V in HU/V . If xV is an element of
UH/U such that Z/V = (Z/V )xV , then (Z/V )xV = (ZxV )/V = Zx/V
implies Zx = Z. Since V H ≤ Z, the subgroup Z is self – normalizing,
in particular, x ∈ Z. In other words, every subgroup of UH/V , which
is intermediate for HV/V , is self – normalizing. By Lemma 1.5 HV/V
is abnormal in HU/V . In turn it follows that HV is abnormal in HU .
By the induction hypothesis H is abnormal in V H. The application of
Lemma 1.1 gives that H is abnormal in HU = HAα. Since it is valid for
each ordinal α, H is abnormal in HAγ = G.
Corollary 1.7. Let G be a hyperabelian group and let H be a subgroup of
G. Then H is abnormal in G if and only if every intermediate subgroup
for H coincides with its normalizer in G.
Corollary 1.8. Let G be a soluble group and let H be a subgroup of G.
Then H is abnormal in G if and only if every intermediate subgroup for
H coincides with its normalizer in G.
For the consideration of artinian – by – X – groups we need some
notions connected with the acting of a group on its abelian subgroup. It
is better to formulate these concepts using the Modules Theory language.
Let R be a ring, G be a group, A be an RG- module, B, C be RG -
submodules of A such that B ≤ C. Factor C/B is called central (more
exactly RG – central) (respectively eccentric or RG – eccentric) if G =
CG(C/B) (respectively G 6= CG(C/B)). Put
ζRG(A) = {a ∈ A|aRG is RG-central}.
76 Abnormal subgroups and Carter subgroups...
Clearly ζRG(A) is an RG – submodule of A. This submodule is called
the RG – center of A.
Starting from the RG - center we can construct the upper RG– central
series of the module A:
〈0〉 = A0 ≤ A1 ≤ ...Aα ≤ Aα+1 ≤ ...Aγ
where A1 = ζRG(A), Aα+1/Aα = ζRG(A/Aα), α < γ, and ζRG(A/Aγ) =
〈0〉.
The last term Aγ of this series is called the upper RG – hypercenter
of the module A and denote by ζ∞RG(A). If A = Aγ , then the module A is
called RG – hypercentral; if γ is finite, then A is called RG – nilpotent.
On the other hand, an RG – submodule C of A is said to be RG -
hypereccentric, if it has an ascending series
〈0〉 = C0 ≤ C1 ≤ ...Cα ≤ Cα+1 ≤ ...Cγ = C
of RG – submodules of A such that each factor Cα+1/Cα is an RG –
eccentric simple RG – module, for every α < γ.
Following D.I. Zaitsev [ZD 1], we say that an RG-module A has the
Z - decomposition if the following equality holds A = ζ∞RG(A) ⊕ ε∞RG(A)
where ε∞RG(A) is the maximal RG - hypereccentric RG - submodule of A.
Note that if A is an artinian RG – module, then ε∞RG(A) includes every
RG – hypereccentric RG – submodule, in particular, it is unique. In fact,
let B be an RG – hypereccentric RG – submodule of A, E = ε∞RG(A). If
(B +E)/E is non – zero it includes a non – zero simple RG – submodule
U/E. Since (B + E)/E ∼= B/(B ∩ E), U/E is RG – isomorphic with
some simple RG – factor of B and it follows that G 6= CG(U/E). On
the other hand, (B + E)/E ≤ A/E ∼= ζ∞RG(A), that is G = CG(U/E).
This contradiction proves that B ≤ E. Hence ε∞RG(A) includes every RG
- hypereccentric RG – submodule, in particular, it is unique.
Lemma 1.9. Let G be a group and suppose that A is a normal abelian
subgroup of G such that G/A is hypercentral and A satisfies Min −G. If
ε∞RG(A) 6= 〈1〉 and H is a complement to ε∞RG(A), then H is abnormal in
G.
Proof. By Theorem 1´ of [ZD 1] A has the Z – decomposition A = C×E,
where C = ζ∞ZG(A), E = ε∞RG(A). Factor – group G/E is hypercentral, so
that by Theorem 2 of [ZD 2] G includes a subgroup H such that G = EH
and E ∩ H = 〈1〉. Moreover, every complement to E in G is conjugate
to H. Let S be an intermediate subgroup for H, that is H ≤ S. Then
S = (E∩S)H. Clearly, E∩S is a G – invariant subgroup, moreover, every
L. A. Kurdachenko, I. Ya. Subbotin 77
S – invariant (even H – invariant) subgroup of E ∩ S is G – invariant.
Put L = NG(S). Similarly L = (E∩L)H. If L 6= S, then E∩S 6= E∩L.
In factor – group L/(E ∩ S) we have S/(E ∩ S) = H(E ∩ S)/(E ∩ S).
If x ∈ (L ∩ E)\S, then [x, S] ≤ S, so that [x, S](E ∩ S)/(E ∩ S) ≤
H(E∩S)/(E∩S). On the other hand, [x, S] ≤ E, that is [x, S] ≤ E∩S.
In other words, factor (E ∩ L)/(E ∩ S) is S – central, and therefore G
– central. However, by the selection of E every G – chief factor of E
is G – eccentric, so that E does not have the G - central factors. This
contradiction shows that S = NG(S). By Corollary 1.7 H is an abnormal
subgroup of G.
The next theorem is about artinian – by – hypercentral groups with
no proper abnormal subgroups.
Theorem 1.10. Let G be a group and suppose that T is a normal sol-
uble subgroup of G such that G/T is hypercentral and T satisfies Min
– G. Group G is hypercentral if and only if G has no proper abnormal
subgroups.
Proof. Let
〈1〉 = T0 ≤ T1 ≤ ... ≤ Td = T
be the derived series of T . We will use induction by d. Let d = 1, that
is A = T1 is abelian. Then A has the Z– decomposition A = C × E,
where C = ζ∞RG(A), E = ε∞RG(A) [ZD 1, Theorem 1´]. Suppose that G
is not hypercentral. It follows that E 6= 〈1〉. Factor – group G/E is
hypercentral, so that by Theorem 2 of [ZD 2] G includes a subgroup H
such that G = EH and E ∩ H = 〈1〉. By Lemma 1.9 subgroup H is
abnormal in G. This contradiction shows that E = 〈1〉, and hence G is
hypercentral.
Let now d > 1 and we have already proved that G/T1 is hypercentral.
We can repeat the above arguments and obtain that G is hypercentral.
2. Carter subgroups in some infinite groups
We will consider the existence of Carter subgroups in some artinian – by
– hypercentral groups.
Note that if G is an artinian – by – hypercentral group, then its hyper-
central residual R coincides with locally nilpotent residual K, moreover
G/K is hypercentral. Indeed, K ≤ R. Since G/K is locally nilpotent,
it has a central series Z (see, for example, [RD 2, Corollary to Theorem
78 Abnormal subgroups and Carter subgroups...
8.24]). Put Z1 = {Z ∩ (R/K)|Z ∈ Z}. Since R/K satisfies Min – G, it
has a minimal element C/K. But every chief factor of a locally nilpotent
group is central (see, for example, [RD 1, Corollary 1 to Theorem 5.27]).
In other words, ζ(G/K) is non – identity. Using the similar arguments
and transfinite induction, we obtain that R/K has an ascending G –
central series. Since G/R is hypercentral, G/K is likewise hypercentral.
Theorem 2.1. Let G be an artinian – by – hypercentral group and
suppose that its locally nilpotent residual K is nilpotent. Then G in-
cludes a minimal abnormal subgroup L. Moreover, L is a maximal hyper-
central subgroup and it includes the upper hypercenter of G. In particular,
G = KL. If H is another minimal abnormal subgroup, then H is conju-
gate to L.
Proof. Let
〈1〉 = K0 ≤ K1 ≤ ... ≤ Kc = K
be the upper central series of K. We will use induction by c. Let c = 1,
that is A = K1 is abelian. Then A has the Z – decomposition A =
C × E, where C = ζ∞RG(A), E = ε∞RG(A) [ZD 1, Theorem 1´]. Factor –
group G/E is hypercentral, so that by Theorem 2 of [ZD 2] G includes
a subgroup L such that G = EL and E ∩ L = 〈1〉. By isomorphism
L ∼= G/E, subgroup L is hypercentral. By Lemma 1.9 L is abnormal
in G. Let S be an intermediate subgroup for L. Then S = (E ∩ S)L.
Since E is abelian, the equation G = ES implies that every S – invariant
subgroup of E is also G – invariant. In particular, E∩S satisfies Min – G.
Every G – chief factor of E is G – eccentric. Thus every S – chief factor of
E∩S is S – eccentric. It follows that if S 6= L (that is E∩S 6= 〈1〉), then S
can not be a hypercentral subgroup. Let H be another minimal abnormal
subgroup of G. Then HE/E is an abnormal subgroup of G/E. Since a
hypercentral group does not include proper abnormal subgroups [KUS],
HE/E = G/E or HE = G. Suppose that E ∩ H 6= 〈1〉. Again each H -
invariant subgroup of E is also G – invariant. In particular, E∩H satisfies
Min –H and E ∩ H = ε∞RG(E ∩ H). By Theorem 2 of [ZD 2] H includes
a subgroup U such that H = (E ∩ H)U and (E ∩ H) ∩ U = 〈1〉. Since
E ∩ H 6= 〈1〉, H 6= U . By Lemma 1.9 U is abnormal in H. Theorem 1.2
yields that U is abnormal in G. However, this contradicts the selection
of H. This contradiction shows that E ∩H = 〈1〉. By Theorem 2 of [ZD
2] H is conjugate to subgroup L.
Suppose now that c > 1 and we have already proved theorem for
factor – group G/A. Let V/A be a minimal abnormal subgroup of G/A.
Since CG(A) ≥ K, G/CG(A) is hypercentral. Then A has the Z – decom-
position A = C × E, where C = ζ∞RG(A), E = ε∞RG(A) [ZD 1, Theorem
L. A. Kurdachenko, I. Ya. Subbotin 79
1´]. The equation G/A = (T/A)(V/A) and the inclusion A ≤ ζ(T ) imply
that every V – invariant subgroup of A is G – invariant. In other words,
A satisfies Min – V and C = ζ∞RG(A), E = ε∞RG(A). Factor – group V/E
is hypercentral, so that by Theorem 2 of [ZD 2] V includes a subgroup
L such that V = EL and E ∩ L = 〈1〉. By isomorphism L ∼= V/E, L is
hypercentral. As we proved above, L is abnormal in V , and Theorem 1.2
yields that L is abnormal in G because V is abnormal in G by induction
hypothesis. Since L is hypercentral, L is a minimal abnormal subgroup
of G. Let H be another minimal abnormal subgroup of G. Since G/T is
hypercentral, HT = G. The inclusion T1 ≥ ζ(T ) implies that every H –
invariant subgroup of T1 is also G – invariant. In particular, T1 satisfies
Min – H. By the same reason, every factor Tj/Ti−1 satisfies Min – H
for each j, 1 ≤ j ≤ c. It follows that T satisfies Min –H, in particular,
H ∩ T satisfies Min – H. By isomorphism H/(H ∩ T ) ∼= HT/T factor
– group H/(H ∩ T ) is hypercentral. If W is an abnormal subgroup of
H, then by Theorem 1.2 W is abnormal in G. By the selection of H it
follows that H does not include a proper abnormal (in H) subgroup. By
Theorem 1.10 H is hypercentral. So HA/A is an abnormal hypercentral
subgroup of G/A. On other words, HA/A is a minimal abnormal sub-
group of G/A. By inductive hypothesis there is an element g ∈ G such
that (LA/A)gA = HA/A. Then H ≤ LgA. Since every H – invariant
subgroup of A is also G – invariant, A satisfies Min – H and C = ζ∞RG(A),
E = ε∞RG(A). In particular, HC is hypercentral. Since H is abnormal
in HC, it follows that HC = H. Furthermore, H ∩ E = 〈1〉, so H is
a complement to E in HA = LgA. By Theorem 2 of [ZD 2] there is an
element z ∈ HA such that H = Lgz.
Corollary 2.2. Let G be an artinian – by – hypercentral group and sup-
pose that its locally nilpotent residual K is nilpotent. Then G includes
a hypercentral abnormal subgroup L. Moreover, L is maximal hyper-
central subgroup and it includes the upper hypercenter of G. In particular,
G = KL. If H is another hypercentral abnormal subgroup, then H is con-
jugate to L.
By Theorem 2.1 a minimal abnormal subgroup is hypercentral. Con-
versely, if L is an abnormal hypercentral subgroup then L does not in-
clude a proper abnormal subgroup [KUS]. This means that L is a minimal
abnormal subgroup.
Theorem 2.3. Let G be an artinian – by – hypercentral group and sup-
pose that its locally nilpotent residual K is nilpotent. If L is a Carter
subgroup of G, then L is a LN – covering subgroup of G. Conversely, if
H is a LN – covering subgroup of G, then H is a Carter subgroup of G.
80 Abnormal subgroups and Carter subgroups...
Proof. Let L be a Carter subgroup of G (its existence follows from The-
orem 2.1). If L ≤ S, then L is an abnormal subgroup of S. Let
〈1〉 = K0 ≤ K1 ≤ ... ≤ Kc = K
be the upper central series of T . By Theorem 2.1 G = LK. It follows that
every L – invariant subgroup of Kj/Kj−1 is G – invariant, 1 ≤ j ≤ c. In
particular, Kj/Kj−1 satisfies Min – L, 1 ≤ j ≤ c. Put Qj = S∩Kj , 0 ≤
j ≤ c. Then every factor Qj/Qj−1 is central in S ∩K and satisfies Min –
L. It follows that S∩K is a nilpotent subgroup satisfying Min – L. Since
S/(S∩K) ∼= SK/K is hypercentral and L(S∩K)/(S∩K) is its abnormal
subgroup, L(S ∩K) = S, because a hypercentral group does not include
a proper abnormal subgroup. It follows that S ∩ K satisfies Min –S. In
other words, S is an artinian – by – hypercentral group. Let R be the
locally nilpotent residual of S. As we have already noted prior to this
theorem, S/R is hypercentral. Since LR/R is an abnormal subgroup of
S/R, LR = S. This means that L is a LN – covering subgroup of G.
Conversely, let H be an arbitrary LN – covering subgroup of G. We
will prove that H is an abnormal hypercentral subgroup of G. We will
use for this induction by c. Let first c = 1, that is A = K1 is abelian.
Then A has the Z – decomposition A = C × E, where C = ζ∞RG(A),
E = ε∞RG(A) [ZD 1, Theorem 1´]. Clearly, E is the hypercentral residual
(and the locally nilpotent residual) of G, so that EH = G. Suppose that
B = E ∩ H 6= 〈1〉. Obviously, B is a G – invariant subgroup of E. Since
E satisfies Min – G, B includes a minimal G – invariant subgroup M .
The equation G = EH yields that every H – invariant subgroup of E
is likewise G – invariant. It follows that M is a minimal H – invariant
subgroup. However every chief factor of a locally nilpotent group H is
central ( see, for example, [RD 1, Corollary 1 to Theorem 5.27]), so that
M is H - central. In this case M is central in G. This contradicts the
inclusion M ∩ E = ε∞RG(A). This contradiction shows that H ∩E = 〈1〉.
Then H ∼= HE/H = G/E is hypercentral. By Lemma 1.9 H is abnormal
in G.
Suppose now that c > 1 and consider factor – group G/A. It is not
hard to see that HA/A is a LN – covering subgroup of G/A. By induction
hypothesis HA/A is an abnormal hypercentral subgroup. Consider now
a subgroup HA. Clearly H is a LN – covering subgroup of HA. We have
already proved that H is a hypercentral subgroup and H is abnormal in
HA. By Lemma 1.1 H is abnormal in G. Hence H is a Carter subgroup
of G.
Theorem 2.4. Let G be an artinian – by – hypercentral group and sup-
pose that its locally nilpotent residual K is nilpotent. If L is a Carter
L. A. Kurdachenko, I. Ya. Subbotin 81
subgroup of G, then L is a locally nilpotent projector of G. Conversely,
if D is a locally nilpotent projector of G, then H is a Carter subgroup of
G.
Proof. Let L be a Carter subgroup of G; its existence follows from The-
orem 2.1. If H is a normal subgroup of G, then LH/H is an abnormal
subgroup of G/H. Let K/H be a locally nilpotent subgroup including
LH/H. Since a locally nilpotent group does not include a proper abnor-
mal subgroups [KUS], LH/H = K/H. This means that L is a locally
nilpotent projector.
Conversely, let D be an arbitrary locally nilpotent projector of G.
Since DK/K is a maximal locally nilpotent subgroup of a hypercentral
group G/E, DK = G. Let
〈1〉 = K0 ≤ K1 ≤ ... ≤ Kc = K
be the upper central series of T . We will prove that D is an abnormal
hypercentral subgroup of G. We will use for this the induction by c. Let
first c = 1, that is A = K1 is abelian. Then A has the Z – decomposition
A = C ×E, where C = ζ∞RG(A), E = ε∞RG(A) [ZD 1, Theorem 1´]. Since
E is a hypercentral residual of G, by above DE = G. As in the proof
of Theorem 2.3 we can prove that E ∩ D = 〈1〉. Then D ∼= DE/H =
G/E is hypercentral. By Lemma 1.9 D is abnormal in G. Suppose now
that c > 1 and consider factor – group G/A. Obviously DA/A is a
locally nilpotent projector of G/A. By induction hypothesis DA/A is an
abnormal hypercentral subgroup. Consider now a subgroup HA. The
inclusion A ≤ ζ(K) and the equation G = KD implies that every D
– invariant subgroup of A is G – invariant. In particular, A satisfies
Min – G. Since DA/A is hypercentral, A has the Z – decomposition
A = Z × U , where C = ζ∞RD(A), U = ε∞RD(A) [ZD 1, Theorem 1’].
Moreover, C = ζ∞RG(A) = ζ∞RD(A), ε∞RG(A) = ε∞RD(A). Since D is a
maximal locally nilpotent subgroup, Z ≤ D. Using the arguments of the
proof of Theorem 2.3 and the equation ε∞RG(A) = ε∞RD(A) we can prove
that D ∩ U = 〈1〉. Lemma 1.9 implies that D is abnormal in DA. By
Lemma 1.1 D is abnormal in G. Hence H is a Carter subgroup of G.
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Contact information
Leonid A.
Kurdachenko
Mathematics Department, National Dne-
propetrovsk University, Provylok Naukovyi
13, 49050 Dnepropetrovsk, Ukraine
E-Mail: mmf@ff.dsu.dp.ua
Igor Ya. Subbotin Mathematics Department, National Univer-
sity, 9920 S. La Cienega Blvd. Inglewood,
CA 90301, USA
E-Mail: isubboti@nu.edu
Received by the editors: 08.10.2004
and in final form 24.03.2005.
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