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The objective of this research paper is to study numerical relationships 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(<i>D</i>)- block and give a necessary and sufficient condition of a block with a cyclic defect group to be a k(<i>D</i>) -block in term of its inertial index. We believe that the notion and the results in this work will contribute to the developments of the theory of blocks of finite groups.

Let

This result led us to think about numerical relationships between a

For a bound of

See [

However, we have arisen a question about which blocks and which conditions ensure the equality

as well as some examples for small

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

find that

Since

Definition 1.1 Let

number of ordinary irreducible characters of

Let us in the following definition consider an equality mod

Definition 1.2 Let

number of ordinary irreducible characters of

It is clear that a strongly

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

At the end of the paper, we use the computations and the results in [

We shall start with some examples which illustrate the phenomenon of

Example 1.3 For

group of the principal

conclusion holds for the principal

and

Example 1.4 Let

Example 1.5 Let

Example 1.6 For

quaternion group

Example 1.7 Now, we have faced the first example which does not obey our speculation. It is the first non

abelian simple group:

that for

Example 1.8 The principal 3-block for

1) Let

2) It is well known that if

3) We know that if

4) For

In this section, we discuss

the inertial index of

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 [

Lemma 2.1 Let

With the above notation, we characterize strongly

Theorem 2.2 Let

Proof: Assuming that

Then we have

Remark 2.3 We get an analogue result of Theorem 2.2 for

solving the congruency equation

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

Lemma 3.1 Let

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

Lemma 3.2 Let

We would like to mention that the origin of the concept of block theory is due to Brauer (see [

Theorem 3.3 Let

Proof: Using Lemma 3.1, we have that every ordinary irreducible character of

We would like to thank the anonymous referees for providing us with constructive comments and suggestions.