On k(D)-blocks

The objective of this research paper is to study the relationship between a block of a finite group and a defect group of such block. We define a new notion which is called a strongly $k(D)$-block and give a necessary and sufficient condition of a block with a cyclic defect group to be a $k(D)$-block in term of its inertial index.


Introduction
Let p be a prime number and B a p-block of a finite group G with a defect group D of order p d . In [1], we proved that k(B) ≡ k(D) Mod(p), where k(B) is the number of the ordinary irreducible characters belonging to B and k(D) is the number of the ordinary irreducible characters of the defect group D, which is an extra special p-group of order p 3 and exponent p, for an odd prime number p. 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 p-block and its defect group. In the present work, we are free from any condition about the prime number p. A question about the existence of a function on the natural numbers which relates some block-invariants under consideration is known as Brauer's 21-problem (see [2,13]). In fact, Brauer asked whether it is the case that k(B) ≤ |D| in general. This question is known as Brauer k(B)-conjecture.
However, we have arisen a question about which blocks and which conditions ensure the equality k(B) = k(D) as well as ensure the congruency k(B) ≡ k(D) Mod(p). We have studied some general cases as well as some examples for small G. Then we try to characterize such blocks which have cyclic defect groups in the term 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 which we have already considered satisfy the equality k(B) = k(D). However, for G = A 5 with p = 5, we find that k(B 0 (G)) = 4, where B 0 (G) is the principal 5-block of G. But k(D) = 5, where D ∈ Syl 5 (G). Since A 5 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, p-blocks satisfy either the equality or the congruency relation. We call those groups which do not satisfy our congruency relation exotic groups. We can consider an equality mod the prime number p as in the following definition. It is clear that a strongly k(D)-block is a k(D)-block. However, we shall see in Example 1.3 some k(D)-blocks which are not strongly k(D)-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 k(D) is the number of the conjugacy classes of D. 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 k(D)-blocks with cyclic defect groups. Our main tool is Dade theorem for the number of irreducible characters of a block with a cyclic defect group; ( see [3] and [5, Page 420]).
In the end of the paper, we use the computations and the results in [9,10] to see that such phenomena do occur quite often in block theory.

General cases for the notion of k(D)-blocks
1. Let p be a prime number and G a finite group. Assume that the prime number p does not divide the order of the group G. Then each block of G has defect zero, (see Theorem 6.29 Page 247 in [11] ). Hence such block is a strongly k(D)-block.
2. It is well known, see Problem 13 in Chapter 5 page 389 in [11] that if G = DO p ′ (G) is a p-nilpotent group with a Sylow p-subgroup D and the maximal normal p ′ -subgroup of G then k(B) = k(D). Certainly, a nilpotent p-block is a strongly k(D)-block. With the above notation, we characterize strongly k(D)-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 k(D)-blocks.

The interplay with fundamental results
There are fundamental progress in solving Brauer problems. We recast the following result which is due to R. Kessar and G. Malle [10,HZC1]. This result can be used to see an strongly block B with abelian defect group D of order p d as such that k 0 (B) = k(D), where k 0 (B) is the number of ordinary irreducible characters of height zero belonging to B. Let us conclude this paper by mentioning the following lemma in such a way that we rely on the computation in [9, Proposition 2.1] by B. Kulshammer and D. Sambale. These computations guarantee that the phenomena of strongly k(D)-block occur quite often in the theory of blocks. In fact, Lemma 3.2 can be replaced by the following much stronger result. Theorem 3.3 Let G be a finite group and p = 2 or 3. Then each p-block of G with abelian defect group is a k(D)-block.