Camina groups with few conjugacy classes
Let G be a finite group having a proper normal subgroup K such that the conjugacy classes outside K coincide with the cosets of K. The subgroup K turns out to be the derived subgroup of G, so the group G is either abelian or Camina. Hence, we propose to classify Camina groups according to the number...
Gespeichert in:
| Datum: | 2010 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2010
|
| Schriftenreihe: | Algebra and Discrete Mathematics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/154601 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Camina groups with few conjugacy classes / L. Cangelmi, A.S. Muktibodh // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 38–47. — Бібліогр.: 8 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-154601 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1546012019-06-16T01:32:29Z Camina groups with few conjugacy classes Cangelmi, L. Muktibodh, A.S. Let G be a finite group having a proper normal subgroup K such that the conjugacy classes outside K coincide with the cosets of K. The subgroup K turns out to be the derived subgroup of G, so the group G is either abelian or Camina. Hence, we propose to classify Camina groups according to the number of conjugacy classes contained in the derived subgroup. We give the complete characterization of Camina groups when the derived subgroup is made up of two or three conjugacy classes, showing that such groups are all Frobenius or extra-special. 2010 Article Camina groups with few conjugacy classes / L. Cangelmi, A.S. Muktibodh // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 38–47. — Бібліогр.: 8 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20D25; 20E45. http://dspace.nbuv.gov.ua/handle/123456789/154601 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Let G be a finite group having a proper normal subgroup K such that the conjugacy classes outside K coincide with the cosets of K. The subgroup K turns out to be the derived subgroup of G, so the group G is either abelian or Camina. Hence, we propose to classify Camina groups according to the number of conjugacy classes contained in the derived subgroup. We give the complete characterization of Camina groups when the derived subgroup is made up of two or three conjugacy classes, showing that such groups are all Frobenius or extra-special. |
| format |
Article |
| author |
Cangelmi, L. Muktibodh, A.S. |
| spellingShingle |
Cangelmi, L. Muktibodh, A.S. Camina groups with few conjugacy classes Algebra and Discrete Mathematics |
| author_facet |
Cangelmi, L. Muktibodh, A.S. |
| author_sort |
Cangelmi, L. |
| title |
Camina groups with few conjugacy classes |
| title_short |
Camina groups with few conjugacy classes |
| title_full |
Camina groups with few conjugacy classes |
| title_fullStr |
Camina groups with few conjugacy classes |
| title_full_unstemmed |
Camina groups with few conjugacy classes |
| title_sort |
camina groups with few conjugacy classes |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/154601 |
| citation_txt |
Camina groups with few conjugacy classes / L. Cangelmi, A.S. Muktibodh // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 38–47. — Бібліогр.: 8 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT cangelmil caminagroupswithfewconjugacyclasses AT muktibodhas caminagroupswithfewconjugacyclasses |
| first_indexed |
2025-07-14T06:39:03Z |
| last_indexed |
2025-07-14T06:39:03Z |
| _version_ |
1837603377257644032 |
| fulltext |
A
D
M
D
R
A
F
T
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 9 (2010). Number 2. pp. 38 – 47
c© Journal “Algebra and Discrete Mathematics”
Camina groups with few conjugacy classes
Leonardo Cangelmi and Arun S. Muktibodh
Communicated by Sushchansky V.I.
Abstract. Let G be a finite group having a proper normal
subgroup K such that the conjugacy classes outside K coincide
with the cosets of K. The subgroup K turns out to be the derived
subgroup of G, so the group G is either abelian or Camina. Hence,
we propose to classify Camina groups according to the number
of conjugacy classes contained in the derived subgroup. We give
the complete characterization of Camina groups when the derived
subgroup is made up of two or three conjugacy classes, showing that
such groups are all Frobenius or extra–special.
Introduction
Let G be a finite group, and K a proper normal subgroup of G. Then there
are two natural partitions of G. One partition is given by the conjugacy
classes, while the other one by the cosets of K. We assume that these two
partitions coincide in G \K. In other words, we assume that cl(x) = xK
for all x ∈ G \ K. This notion was introduced and investigated by the
second author in [5], where he proposed to call such a group Con–Cos,
outlining the coincidence of conjugacy classes with cosets, apart from
those contained in K.
The subgroup K turns out to be the derived subgroup of G, so that
G is a Camina group, unless K = 1, and in such a case the group G is
abelian. Camina groups were introduced by A.R. Camina in [1], and they
have been extensively studied by Macdonald, Chillag, Mann, Scoppola
and Dark. For a complete list of papers on Camina groups, we refer the
reader to [3].
2000 Mathematics Subject Classification: 20D25; 20E45.
Key words and phrases: Camina groups; Frobenius groups; Conjugacy classes.
A
D
M
D
R
A
F
T
L. Cangelmi, A. S. Muktibodh 39
If G is abelian, K = G′ = 1, and there is a complete coincidence of
conjugacy classes and cosets in the whole group. If G is a Camina group
(i.e., a non–abelian Con–Cos group), the Camina kernel K = G′ is a union
of a number of conjugacy classes greater than one. In this paper we study
the cases when K has ‘few’ conjugacy classes, precisely when K is the
union of two or three conjugacy classes. Our main results are the following.
Theorem I. Let G be a finite group. Then, G is a Camina group and G′
is the union of two conjugacy classes if and only if either G is a Frobenius
group with Frobenius kernel Zr
p and Frobenius complement Zpr−1, or G
is an extra–special 2–group.
Theorem II. Let G be a finite group. Then, G is a Camina group and G′
is the union of three conjugacy classes if and only if either G is a Frobenius
group with Frobenius kernel Zr
p and Frobenius complement Z(pr−1)/2, or
G is an extra–special 3–group, or G is Frobenius with Frobenius kernel
Z2
3 and Frobenius complement Q8.
In Section 1 we give the basic properties of Con–Cos groups, explain
the relations with Camina groups, and recall some fundamental facts about
Camina groups. In Sections 2 and 3, we prove, respectively, Theorems I
and II. In Section 4 we give some examples of Camina groups with the
kernel being the union of a number of conjugacy classes greater than three.
Our notation is as follows. We let 1 denote the identity group, Zn the
cyclic group of order n, Dn the dihedral group of order n, and Q8 the
quaternion group of order 8. If x is an element of a group, we let cl(x)
denote the conjugacy class of x. Moreover, when a semidirect product
L⋊M is Frobenius, this is explicitly mentioned.
1. Con–Cos groups and Camina groups
Definition 1.1. A finite group G is Con–Cos when there exists a normal
subgroup K < G such that xK = cl(x), for all x ∈ G \K.
Any abelian group G is Con–Cos, since cl(x) = {x} for all x ∈ G,
and we can take K = 1. Some of the smallest examples of non–abelian
Con–Cos groups are the dihedral group D8 and the quaternion group
Q8 with K = Z2, the dihedral groups D2n, for n odd, with K = Zn, the
alternating group A4 with K = Z2
2 , and the Frobenius group Z5⋊Z4 with
K = Z5.
Lemma 1.2. Let G be a finite group, and K a proper normal subgroup
of G. Then the following conditions are equivalent:
A
D
M
D
R
A
F
T
40 Camina groups with few conjugacy classes
• xK ⊆ cl(x) for all x ∈ G \K;
• K ⊆ {[x, y] : y ∈ G}, for any x ∈ G \K.
Moreover, if the above equivalent conditions hold, then K ⊆ G′.
Proof. Let k ∈ K, and x ∈ G \K. If xk ∈ cl(x), then there exists y ∈ G
such that xk = y−1xy, hence k = [x, y]. Conversely, if there exists y ∈ G
such that k = [x, y], then xk ∈ cl(x). Moreover, if the equivalent conditions
hold for G, then for any x ∈ G \K, the set {[x, y] : y ∈ G} is obviously
contained in G′, so that K ⊆ G′.
Remark 1.3. The condition K ⊆ G′ does not imply that the two
equivalent conditions of Lemma 1.2 hold true. For instance, consider
the group G = Z3 ⋊ Z4 = 〈a, t : a3 = t4 = 1, t−1at = a2〉 and take
K = G′ = {1, a, a2}. Then, t2 ∈ Z(G), hence t2K 6⊆ cl(t2).
Lemma 1.4. Let G be a finite group, and K a proper normal subgroup
of G. Then the following conditions are equivalent:
• cl(x) ⊆ xK, for all x ∈ G;
• G′ ⊆ K.
Proof. Assume the first condition, and consider the factor group G/K.
Let x, y ∈ G. Then (yK)−1(xK)(yK) = (y−1xy)K ⊆ xK, so that G/K
is abelian, and G′ ⊆ K. Conversely, assume the second condition, and let
x ∈ G. For any y ∈ G, we have [x, y] ∈ K, hence y−1xy ∈ xK.
Remark 1.5. The inclusion G′ ⊆ K is not generally an equality. Consider
the example of D8 with K = Z4.
Proposition 1.6. Let G be a Con–Cos group, and let K be a proper
normal subgroup of G satisfying the condition of Definition 1.1. Then
K = G′, and G′ = {[x, y] : y ∈ G}, for any x ∈ G \G′. Therefore, a group
G is Con–Cos if and only if G′ < G and cl(x) = xG′ for all x ∈ G \G′.
Proof. It is a direct consequence of Lemmas 1.2 and 1.4.
Remark 1.7. It is not true that K = G′ implies cl(x) = xK for all
x ∈ G \K. A counterexample is the group Z3 ⋊ Z4 given above.
The previous proposition shows that if G is a non–abelian Con–Cos
group, then G is a Camina group. Indeed, a Camina pair is a pair (G,K),
where G is a finite group and K is a subgroup of G such that 1 < K < G
and the equivalent conditions of Lemma 1.2 hold. Such a subgroup K is
called Camina kernel. Moreover, a finite group G is a Camina group when
A
D
M
D
R
A
F
T
L. Cangelmi, A. S. Muktibodh 41
(G,G′) is a Camina pair (see [1] and [3]). Therefore, by Proposition 1.6, a
Camina group is Con–Cos, and conversely a non–abelian Con–Cos group
is a Camina group. We complete these claims in the following proposition.
Proposition 1.8. Let G be a finite group. The following conditions are
equivalent:
• G is Con–Cos;
• G is either a Camina group, or an abelian group;
• xG′ = cl(x), for all x ∈ G \G′;
• G′ = {[x, y] : y ∈ G}, for any x ∈ G \G′.
Therefore, our study of Con–Cos groups converts to that of Camina
groups.
Camina proved that if (G,K) is a Camina pair, then one of the following
three conditions holds (see [1]):
• G is Frobenius with Frobenius kernel K
• K is a p–group, for some prime p
• G/K is a p–group, for some prime p.
A well known result about Camina pairs and Camina groups is about
the center subgroup. We prove here again, since it is of fundamental
importance for the proofs in the sequel.
Lemma 1.9. Let (G,K) be a Camina pair. Then, Z(G) ⊆ K. In partic-
ular, if G is Camina, then Z(G) ⊆ G′.
Proof. Let z ∈ Z(G) \ 1. If z /∈ K, then we have zK ⊆ cl(z) = {z}. This
would imply |K| = 1, which is not possible.
A wide subclass of Camina groups is given by Frobenius groups with
abelian complement. Indeed, if G = N ⋊ H is a Frobenius group with
abelian complement, the Frobenius complement H has to be cyclic, by
the structure theorem on Frobenius groups (see, e.g., Section 10.5 of [6]).
Proposition 1.10. Let G be a Frobenius group with Frobenius kernel N
and cyclic Frobenius complement H. Then, G is a Camina group with N
as Camina kernel and has trivial center.
A
D
M
D
R
A
F
T
42 Camina groups with few conjugacy classes
Proof. Note first that G′ ⊆ N , since G/N is abelian. Let h ∈ H \ 1.
Then cl(h) ⊆ hN , by Lemma 1.4. On the other hand, CG(h) = H, since
the center of G is trivial. Therefore, |cl(h)| = |G|/|H| = |N |, so that
cl(h) = hN . The same occurs for any other x ∈ G \ N , since we can
replace H with its conjugate subgroup containing x.
Another wide subclass of Camina groups is given by extra–special
p–groups (for a complete description of such groups, see Section 5.3 of
[6]).
Proposition 1.11. Let G be an extra–special p–group, with p prime. Then,
G is a Camina group with Camina kernel equal to the center subgroup.
Proof. Let K = G′. Since G is an extra–special p–group, we have K =
Z(G) = Zp. Then |K| = p, and K contains exactly p conjugacy classes.
For any x ∈ G \K, we have cl(x) ⊆ xK, by Lemma 1.4. On the other
hand, it is easy to see that if |G| = p2n+1 then |CG(x)| = p2n. Hence
|cl(x)| = p, and cl(x) = xK.
Let G be a Con–Cos group, that is an abelian group or a Camina
group. Consider the number n of conjugacy classes contained in K = G′.
Then, n = 1 if and only if G is abelian. Otherwise, G is a Camina group,
and we propose the following definition.
Definition 1.12. A n–Camina group, where n ≥ 2, is a Camina group
G with G′ being the union of n conjugacy classes.
Taking into account this definition, we can briefly say that Theorems
I and II give the classification, respectively, of 2–Camina and 3–Camina
groups.
2. Camina kernels with two conjugacy classes
Let G be a 2–Camina group, so that G′ = 1 ∪ cl(a), for some a ∈ G with
a /∈ 1. Then, it is easy to see that the subgroup G′ is an abelian elementary
group. On the other hand, we remark that groups with a normal subgroup
which is the union of two conjugacy classes have been thoroughly studied
by Shahryari and Shahabi in [8].
We separate the proof of Theorem I in the following two propositions,
according to whether the center of G is trivial or not.
Proposition 2.1. Let G be a 2–Camina group with Z(G) = 1. Then,
G is a Frobenius group Zr
p ⋊ Zpr−1, for some prime p and some r ≥ 1.
Conversely, such Frobenius groups are 2–Camina.
A
D
M
D
R
A
F
T
L. Cangelmi, A. S. Muktibodh 43
Proof. Since G′ is the unique minimal normal subgroup of G, by Theorem
2.1 (a) of [8], or by Lemma 12.4 of [4], G is Frobenius with elementary
abelian kernel G′. Moreover, the complement H is abelian, since it is
isomorphic to G/G′, hence it is cyclic. Therefore, G′ = Zr
p for some prime
p and some r ≥ 1, and H = Zh for some h ≥ 2. As G′ \ 1 is a single
conjugacy class, the action of H on G′ by conjugation has to be transitive,
so that |H| = pr − 1.
Conversely, it is easy to see that Frobenius group Zr
p ⋊ Zpr−1 are
2–Camina.
Proposition 2.2. Let G be 2–Camina and Z(G) 6= 1. Then G′ = Z(G) =
Z2, and G is an extra–special 2–group. Conversely, any extra–special 2–
group is a 2–Camina group.
Proof. Let a ∈ Z(G) \ 1. Then a ∈ G′, by Lemma 1.9, and cl(a) = {a}.
Hence, G′ = {1, a}, and G′ = Z(G) = Z2. Now, let B = G/G′. We have
just to show thatB is a 2–group. Let x, y ∈ G. We have [x, y2] = [x, y]2 = 1,
so that y2 commutes with all the elements of G (see also [6]). Hence
y2 ∈ Z(G) = G′, for all y ∈ G.
Conversely, a straight verification shows that any extra–special 2–group
is 2–Camina.
3. Camina kernels with three conjugacy classes
The groups with a normal subgroup which is the union of three conjugacy
classes have been deeply studied by Shahryari and Shahabi in [7]. In the
following proposition, we extract the relevant results we need to our aims.
Proposition 3.1. Let G be a group with a normal subgroup K which is
the union of three conjugacy classes. Then, there are only three possible
cases:
1. K is an abelian elementary group of odd order, i.e. K = Zr
p , for
some odd prime p and some integer r ≥ 2, the two conjugacy classes
different from 1 have the same size, and K is a minimal normal
subgroup of G, or
2. K is a metabelian p–group, i.e. K ′′ = 1, and K ′ is elementary
abelian, K ′ = Zs
p, for some prime p and some integer s ≥ 1, or
3. K is Frobenius with Frobenius kernel K ′, which is elementary abelian,
K ′ = Zs
p, and Frobenius complement of prime order, Zq, for some
primes p and q and some integer s ≥ 1, with q | ps − 1.
A
D
M
D
R
A
F
T
44 Camina groups with few conjugacy classes
Proof. See Cases A, B, and C of [7].
We first consider the possible order of the center of a 3–Camina group.
Lemma 3.2. Let G be a 3–Camina group. Then |Z(G)| ≤ 3, and if
|Z(G)| = 3 then G′ = Z(G).
Proof. By Lemma 1.9, Z(G) ⊆ G′. If |Z(G)| = 3, then G′ contains
the three conjugacy classes cl(x), for x ∈ Z(G), hence G′ = Z(G). If
|Z(G)| > 3, then G′ would contain more than three conjugacy classes,
which is not possible.
We now prove Theorem II, considering separately the three possible
cases of Proposition 3.1.
Proposition 3.3. Let G be a 3–Camina group, and assume that case (1)
of Proposition 3.1 holds for K = G′. If Z(G) = 1, then G is a Frobenius
group Zr
p ⋊ Z(pr−1)/2, for some odd prime p and some r ≥ 1; whereas, if
Z(G) 6= 1, then G is an extra–special 3–group. Conversely, such Frobenius
groups and extra–special 3–groups are 3–Camina groups.
Proof. When Z(G) = 1, the derived subgroup G′ is necessarily the unique
minimal normal subgroup of G. Hence, by Lemma (12.3) of [4], G is
Frobenius with Frobenius kernel G′ and cyclic Frobenius complement.
Since G′ \ 1 contains exactly two conjugacy class of the same size, the
order of the complement has to be (pr − 1)/2. Also, this implies that p is
odd.
When the center is not trivial, it is elementary abelian, since G′ is
such. Since its order is odd, by Lemma 3.2 we deduce Z(G) = Z3, and
G′ = Z(G). Using the same arguments of Proposition 2.2, it follows at
once that G is an extra–special 3–group.
The converse claim is easy to verify.
Before dealing with the cases (2) and (3) of Proposition 3.1, we prove
a lemma holding in both cases.
Lemma 3.4. Let G be a 3–Camina group, with K = G′ not abelian. Then
K ′ is a minimal normal subgroup of G, and G/K ′ is a 2–Camina group.
Hence there are only two possible cases:
1. G/K ′ is a Frobenius group Zt
r ⋊ Zrt−1, for some prime r and some
positive integer t, or
2. G/K ′ is an extra–special 2–group.
A
D
M
D
R
A
F
T
L. Cangelmi, A. S. Muktibodh 45
Proof. Since K is a union of three conjugacy classes and K ′ 6= 1, then K ′
is a union of two conjugacy classes, hence it is minimal normal. On the
other hand, G/K ′ is easily seen to be 2–Camina, and the two cases are
the only possible ones for 2–Camina groups, by Theorem I.
Proposition 3.5. There does not exist any 3–Camina group G, such that
case (2) of Proposition 3.1 holds for K = G′.
Proof. Assume that G is 3–Camina, and that case (2) of Proposition
3.1 holds for K = G′. Note that Z(K) ∩K ′ 6= 1, since K is a p–group.
Moreover, Z(K) ∩ K ′ is a normal subgroup of G contained in K ′. By
Lemma 3.4, we have Z(K) ∩ K ′ = K ′, hence K ′ ⊆ Z(K) ⊆ K, which
implies Z(K) = K ′, because K is not abelian and contains exactly three
conjugacy classes. This excludes at once case (2) of Lemma 3.4. Indeed, if
G/K ′ were an extra-special 2–group, then K/K ′ would be a cyclic group
(of order two), and, since K/K ′ = K/Z(K), then K would be abelian.
Therefore, assume that case (1) of Lemma 3.4 holds. Hence, K ′ = Zs
p ,
G/K ′ = Zt
p⋊Zpt−1, and K/K ′ = Zt
p. So, K is a normal Sylow p–subgroup
of G, and G is Frobenius with Frobenius kernel K, by Proposition 1 of
[1]. Then, the Frobenius complement is a cyclic group of order pt − 1. If
p 6= 2, the order of the complement is even, and this implies that K is
abelian, which is not the case. Hence we have to assume that p = 2. So,
G is a Frobenius group K ⋊ Z2t−1, with |K| = 2s+t and |K ′| = 2s. Since
K ′
⋊Z2s−1 is Frobenius too, K ′ = Z(K), and K ′ \ 1 is a single conjugacy
class, then we have s = t.
Therefore, G = K ⋊ Z2s−1, |K| = 22s, K ′ = Zs
2 , and K/K ′ = Zs
2 .
Now, consider the order of the elements in K. All the elements in K ′ \ 1
have order 2, and if all the elements in K \K ′ have also order 2, then K
would be abelian. Hence, there is an element in K \K ′ of order different
from 2. Since K \K ′ is a single conjugacy class, all of its elements have
order 4. Let b ∈ K \K ′. Then cl(b) = K \K ′, so |cl(b)| = 2s(2s − 1), and
|CG(b)| = 2s. Since K ′ = Z(K) ⊆ CG(b), we have CG(b) = K ′, which is a
contradiction, because b ∈ CG(b) \K
′.
Proposition 3.6. Let G be a 3–Camina group, and assume that case (3)
of Proposition 3.1 holds for K = G′. Then G is Frobenius with Frobenius
kernel N and Frobenius complement Q8, where N = Z2
3 , and K = N ⋊Z2.
Conversely, such Frobenius groups are 3–Camina.
Proof. Assume that case (3) of Proposition 3.1 and case (1) of Lemma
3.4 hold. Then, using their notation, we have K/K ′ = Zq = Zt
r, so t = 1
and r = q. Hence, G/K ′ = Zq ⋊ Zq−1, |G| = psq(q − 1) and |K| = psq.
Since p | q − 1, K is a normal Hall subgroup of G, so G is a Frobenius
A
D
M
D
R
A
F
T
46 Camina groups with few conjugacy classes
group with Frobenius kernel K, by Proposition 1 of [1]. Moreover, the
Frobenius complement turns out to be a cyclic group of order q − 1, an
even integer, so that K is abelian, which is false (note that the case q = 2
is not possible, because it would imply K = G).
By Lemma 3.4, the only remaining case to consider is when G/K ′ is
an extra–special 2–group. In this case we have q = 2, so K is a Frobenius
group Zs
p ⋊ Z2 with p 6= 2. On the other hand, since K ′ is a normal Hall
subgroup, G splits over K ′, hence G = Zs
p ⋊H, with H an extra–special
2–group. Note that G/K is a 2–group, G is neither a 2–group (because
Z(G) = 1, being Z(G) ⊆ Z(K) and K Frobenius), nor a Frobenius
group with Frobenius kernel K (otherwise, K would be abelian, being the
Frobenius complement of even order). Therefore, G has property F2(2),
according to [2]. Moreover, H is a Sylow 2–subgroup of class 2. Hence
we can apply Theorem 5.1 of [2], deducing that G is a Frobenius group
with Frobenius kernel of index 2 in K and Frobenius complement H = Q8.
That is, G is the Frobenius group Zs
p⋊Q8. If a ∈ K ′\1, then cl(a) = K ′\1,
hence |cl(a)| = ps − 1. Therefore ps − 1 | 8ps, which implies ps − 1 | 8.
Hence the only possible case is p = 3 and s = 2, which, indeed, gives a
3–Camina group.
4. Camina kernels with more conjugacy classes
A classification of n–Camina groups for a fixed n ≥ 4 seems to be very
difficult to obtain. Nevertheless, here we give some examples of Camina
groups, for which we know the number of conjugacy classes in the kernel.
They are all generalizations of the groups we have encountered in the
previous sections.
Example 4.1. Let G be a Frobenius group with abelian Frobenius kernel
K of order n and cyclic Frobenius complement H of order h. Then G
is a Camina group and the number of conjugacy classes in K = G′ is
(n− 1)/h+ 1.
Indeed, we know that h | n− 1 and, since K is abelian, the conjugacy
classes in K are determined just by the action by conjugation of H . Since
this action is fixed–point–free, for any x ∈ K \ 1 we have |cl(x)| = h.
Example 4.2. Let G be an extra–special p–group. Then, G is a p–Camina
group.
Here the result is almost obvious, since G′ = Z(G) = Zp.
Example 4.3. Let G be a Frobenius group with Frobenius kernel Z2
p ,
with p an odd prime, and Frobenius complement Q8. Then, G is Camina,
A
D
M
D
R
A
F
T
L. Cangelmi, A. S. Muktibodh 47
being K = G′ = Z2
p ⋊ Z2, and the number of conjugacy classes in K is
(p2 − 1)/8 + 2.
Note first that 8 | p2 − 1. If x ∈ Z2
p \ 1, then |cl(x)| = 8. Meanwhile, if
x ∈ K \ Z2
p , we have |cl(x)| = p2. So, in K \ 1 there is a single conjugacy
class containing all the elements of order 2, while the elements of order p
are partitioned in conjugacy classes of size 8. For the structure of such
groups, see also Example 1 of [2].
References
[1] Camina, A.R. (1978), Some conditions which almost characterize Frobenius groups,
Israel J. Math. 31:153–160.
[2] Chillag, D., Macdonald, I.D. (1984), Generalized Frobenius groups, Israel J. Math.
47:111–122.
[3] Dark, R., Scoppola, C.M. (1996), On Camina groups of prime power order, J.
Algebra 181:787–802.
[4] Isaacs, I.M. (1994), Character theory of finite groups (corrected reprint), New York,
Dover.
[5] Muktibodh, A.S. (2003), Generalized Con–Cos groups. In: Advances in Algebra,
Proc. of the ICM satellite conference in algebra and related topics, Hong Kong,
China, August 14–17, 2002; Shum, K.P., et al. Eds.; River Edge, New York, World
Scientific, pp. 434–441.
[6] Robinson, D.J.S. (1996), A Course in the Theory of Groups (GTM 80, second
edition), New York, Springer.
[7] Shahryari, M., Shahabi, M.A. (1998) Subgroups which are the union of three
conjugacy classes, J. Algebra 207:326–332.
[8] Shahryari, M., Shahabi, M.A. (1999), Subgroups which are the union of two conju-
gacy classes, Bull. Iran. Math. Soc. 26:59–71.
Contact information
L. Cangelmi Dipartimento di Scienze
Università di Chieti–Pescara
V.le Pindaro 42
65127 Pescara, Italy
E-Mail: cangelmi@sci.unich.it
A. S. Muktibodh Department of Mathematics
Mohota Science College
Umred Rd.
Nagpur–440009, India
E-Mail: amukti2000@yahoo.com
Received by the editors: 06.03.2009
and in final form ????.
Leonardo Cangelmi and Arun S. Muktibodh
|