1. Introduction
Let
be a prime number and
a
-block of a finite group
with a defect group
of order
. Assuming Dade’s projective conjecture, we prove in [1] that
, where
is the number of the ordinary irreducible characters belonging to
and
, is the number of the ordinary irreducible characters of the defect group
, which is an extra special
-group of order
and exponent
, for an odd prime number
. In other words, we relate the number of the ordinary irreducible characters of a finite group which belong to a certain block and the number of the conjugacy classes of the defect group of that block under consideration.
This result led us to think about numerical relationships between a
-block and its defect group. In the present work, we are free from any condition about the prime number
. A question about the existence of a function on the natural numbers which relates some block-invariants under consideration is known as Brauer
-conjecture (see [2] [3] ). In fact, Brauer asks whether it is the case that
in general.
For a bound of
, it is well known that
for
and
for
.
See [2] -[7] for more details and discussions in this direction.
However, we have arisen a question about which blocks and which conditions ensure the equality
as well as ensure the congruency
. We have studied some general cases
as well as some examples for small
. Then we try to characterize such blocks which have cyclic defect groups in terms of the order of the inertial subgroups.
However, as far as we know, we have not seen a similar relation in the literature. In fact, most of the examples
have already been considered to satisfy the equality
. However, for
with
, we
find that
, where
is the principal 5-block of
. But
, where
.
Since
is a simple group and some unusual properties arise from such group, we think that there are some classes in group theory in such a way.
-blocks satisfy either the equality or the congruency relation.
Definition 1.1 Let
be a prime number,
a finite group and
a
-block of
with defect group
. Write
to mean the number of ordinary irreducible characters of
and write
to mean the
number of ordinary irreducible characters of
. We call
a strongly
-block if
.
Let us in the following definition consider an equality mod
.
Definition 1.2 Let
be a prime number,
a finite group and
a
-block of
with defect group
. Write
to mean the number of ordinary irreducible characters of
and write
to mean the
number of ordinary irreducible characters of
. We call
a
-block if
.
It is clear that a strongly
-block is a
-block. However, we shall see in Example 1.3 some
- blocks which are not strongly
-blocks.
Our main concern is to study finite groups and their blocks which satisfy Definitions 1.1 and 1.2. Note that it is well known that
is the number of the conjugacy classes of
. It is well known that blocks with cyclic defect groups are well understood. This theory is rich and has many applications. So, we shall start by doing some sort of characterization of strongly
-blocks with cyclic defect groups. Our main tool is Dade’s theorem for the number of irreducible characters of a block with a cyclic defect group (see [8] and ([9] , p. 420])).
At the end of the paper, we use the computations and the results in [10] [11] to see that such phenomena do occur quite often in block theory.
1.1. Examples of Strongly
-Blocks,
-Blocks and Non
-Blocks
We shall start with some examples which illustrate the phenomenon of
-blocks.
Example 1.3 For
; the symmetric group of
letters and
, it happens that
, for
and
. However, for any prime number
, the defect
group of the principal
-block of the symmetric group
is an abelain
-group of order
and
. In this situation, we obtain
-blocks which are not strongly
-blocks. A similar
conclusion holds for the principal
-block of the symmetric group
, with
. However, when ![]()
and
, then
, where
.
Example 1.4 Let
be the dihedral group of order
. It has a unique 2-block with five ordinary irreducible characters which obviously coincide with the number of the conjugacy classes of
.
Example 1.5 Let
be the alternating group
. Then for
,
has a unique 2-block with four ordinary irreducible characters which is the same as the number of the conjugacy classes of the defect group. For
, we see that
has one 3-block of defect zero and the principal 3-block with three ordinary irreducible characters with the same number as the number of the conjugacy classes of a Sylow 3-subgroup of
.
Example 1.6 For
; the special linear group, we have for
, the principal
-block has the
quaternion group
as a defect group and indeed,
and
. Then
is a
-block.
Example 1.7 Now, we have faced the first example which does not obey our speculation. It is the first non
abelian simple group:
. Although, for
,
, where
, we observed
that for
,
, where,
. The same obstacle we have faced for the group
, since
when
.
Example 1.8 The principal 3-block for
; the projective special linear group, satisfies
, where
is an extra special
-group of order 27 and exponent 3.
1.2. General cases for the notion of
-blocks
1) Let
be a prime number and
a finite group. Assume that the prime number
does not divide the order of the group
. Then each block of
has defect zero, (see Theorem 6.29 Page 247 in [12] ). Hence such block is a strongly
-block.
2) It is well known that if
is a
-nilpotent group with a Sylow
-subgroup
and the maximal normal
-subgroup of
then
. Certainly, a nilpotent
-block is a strongly
-block (see Problem 13 in Chapter 5 Page 389 in [12] ).
3) We know that if
is a
-group then it has a unique
-block, namely, the principal block and such block is an strongly
-block.
4) For
-blocks with dihedral defect groups, we have
. Hence such blocks are examples of strongly
-blocks.
2.
-blocks with cyclic defect groups
In this section, we discuss
-blocks with cyclic defect groups. Recall that a root of a
-block
of a finite group
with defect group
is a
-block
of the subgroup
such that
(see Chapter 5 Page 348 in [12] ), where
is the centralizer subgroup of
in
. Now for the root
, we define
the inertial index of
to be the natural number
, where
. It is clear that
is a subgroup of
which contains
and the index
is well-defined. The number
above is crucial to investigate some fundamental results in
block theory.
Let us restate the following well known result which was established by Dade regarding the number of irreducible characters in a block with a cyclic defect group. For more detail, the reader can see the proof and other constructions in [8] [13] [14] .
Lemma 2.1 Let
be a
-block of a finite group
with a cyclic defect group
of order
. Then
has
ordinary irreducible characters, where
.
With the above notation, we characterize strongly
-blocks in the term of the inertial index for blocks with cyclic defect groups. Also, we believe that it is worth looking for some positive theorems regarding the notion of
-blocks.
Theorem 2.2 Let
be a
-block of a finite group
with a cyclic defect group
of order
. Then
is a strongly
-block if and only if
or (
and
).
Proof: Assuming that
is a strongly
-block and using Lemma 2.1, we can write
.
Then we have
. Letting
be the variable, we see that the only solution we have is that
or
. The result follows as
divides
. The converse is clear and the main result follows.
Remark 2.3 We get an analogue result of Theorem 2.2 for
-blocks with cyclic defect groups, by
solving the congruency equation
.
3. The interplay with fundamental results
There are fundamental progress in solving Brauer problems. We recast the following result which is due to Kessar and Malle [11, HZC1]. This result can be used to see an strongly block
with abelian defect group
of order
as such that
, where
is the number of ordinary irreducible characters of height zero belonging to
.
Lemma 3.1 Let
be a finite group, and
be a
-block of
with defect group
. If
is abelian, then every ordinary irreducible character of
has height zero.
Let us conclude this paper by mentioning the following lemma in such a way that we rely on the computation in [10, Proposition 2.1] by Kulshammer and Sambale. These computations guarantee that the phenomena of strongly
-block occur quite often in the theory of blocks.
Lemma 3.2 Let
be a finite group, and
be a 2-block of
with a defect group
. If
is an elementary abelian of order 16, then
is a
-block.
We would like to mention that the origin of the concept of block theory is due to Brauer (see [15] -[21] ). For the case for
, see [18] . He dealt with some elements in the center of the defect groups. In our case, we shall assume that the defect groups are abelian groups. In fact, Lemma 3.2 can be replaced by the following much stronger result.
Theorem 3.3 Let
be a finite group and
or 3. Then, each
-block of
with abelian defect group is a
-block.
Proof: Using Lemma 3.1, we have that every ordinary irreducible character of
has height zero. Then, the result is followed by elementary observations in [18] .
Acknowledgements
We would like to thank the anonymous referees for providing us with constructive comments and suggestions.