Received 10 February 2015; accepted 5 March 2015; published 20 March 2015

ABSTRACT

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 - block and give a necessary and sufficient condition of a block with a cyclic defect group to be a -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.

Keywords:

Brauer’s k(B) Problem, Blocks with cyclic defect groups

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.

References