Chinese Journal of Mathematics
Volume 2016 (2016), Article ID 5342926, 7 pages
http://dx.doi.org/10.1155/2016/5342926

Research Article

The Fifth Dimension Subgroup for Metabelian 2 Groups

Shalini Gupta

Department of Mathematics, Punjabi University, Patiala 147002, India

Received 18 April 2016; Accepted 14 June 2016

Academic Editor: Burkhard Külshammer

Copyright © 2016 Shalini Gupta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Given a finite metabelian -group , the object of this paper is to discuss some cases under which . Further, some examples of groups of class , for which but , are discussed.

1. Introduction

Let be the integral group ring of group and let be the augmentation ideal of . For , let be th dimension subgroup and let be th term in lower central series of group . The famous dimension subgroup conjecture states that for all and for all groups . This conjecture has been proved for (see, e.g., [1]). It has been proved in [2] that if is a finite -group, odd prime, then . Keeping in view this result, Rips [3] gave an example of -group for which . Further Tahara [4] gave the structure of and proved that the exponent of is divisible by . It has been shown in [5] that, for a metabelian group, the exponent of is divisible by , and hence for a metabelian -group odd prime . Gupta [6] constructed a -group , generated by four elements, such that . Now, the question arises that whether for a metabelian -group , with at most three generators, or not.

In this paper, we prove that if is metabelian group generated by at most two elements, then (Theorem 2). Further we prove that under certain conditions, a metabelian -group generated by elements satisfies the dimension subgroup conjecture for (Theorem 3). Finally, in Section 3, we give some examples of groups of class , for which .

2. Main Results

We first recall the structure of given by Tahara [4]. Let be a finite group of class 4. Consider the lower central series of . Let be basis of , with as the order of : that is, belongs to . Similarly, let be basis of , with as the order of , and let be basis of , with as the order of . Moreover, these basis elements are chosen in such a way that divides , divides , and divides . Thus, we have

The following theorem gives the structure of .

Theorem 1 (see [4]). With the above notations, is equal to the subgroup generated by the elements

with the following conditions:

Theorem 2. Let be metabelian group with as sum of at most two cyclic groups. Then .

Proof. It is obvious that if metabelian groups are such that is a cyclic group then .
Let be metabelian group and let be sum of two cyclic groups. It is enough to prove the result for a group of class 4. Let , where is a cyclic group generated by with order of and divides . An arbitrary element of is of the form subject to the conditions given in Theorem 1. For and , (9) becomes

Thus,

Now belongs to and belongs to ; thus we have

as for (14) becomes . Now, (16), for , becomes

which implies that

Now,

Using (6) and (7), we get that

Hence, (24) reduces to

Now,

Also (7) implies that

Thus, (28) becomes

as by (7), .
Using (23), (27), and (30) in (21), we get that

Thus, , and hence .

Theorem 3. Let be finite metabelian 2 group and , where are cyclic groups of order . Let , . Then .

Proof. It is enough to prove the result for a group of class 4. Any element of is of the form

Since for , , (13) reduces to

and hence . Now,

Also for , (7) implies that , which gives that

Now for , (12) implies that , which gives that

Thus, (34) reduces to

Hence,

Now, for , (9) becomes

which gives

Equation (14) for becomes

Thus,

Using (16), we get that

Now

Also it can be seen easily from (7) that

Now using (38), (40), (43), (44), and (45), we get that

Hence, .

3. Groups of Class 3

It has been proved that if is a finite -group of class , odd prime, then . Clearly for such a group . Now the question arises that if is group of class 3 with , then is it possible that ? We will discuss some examples of groups of class 3 in which , but .

3.1. Example  1

Let be a group generated by four elements . Set and , , and . Suppose that satisfies the relations

(i), , , and ;(ii), ;(iii);(iv), ;(v);(vi). It has been proved in [6] that is group of class , and . we will show that .

From the above relations we conclude that

(i), , and ;(ii), , , , , and , where denotes the order of an element .

Consider

It is easy to see that , , , , , , , , , , and . Also with the help of (1) and (2), we get that , , , , , and .

Since is a group of class , so an arbitrary element of can be written as

Equation (9) for and becomes

which gives that

Also (17), for , gives that

For and , (18) becomes

Now, (50), (51), and (52) imply that

Again, for and , (9) becomes

Since order of is , combining (53) and (54), we get that .

3.2. Example  2

Let be a group generated by four elements Set , , , and . Suppose that satisfies the following relations:

Consider

It is easy to see that , , , , , , , , and . Also we have , , , , , , , , , , and . It has been proved in [7] that is a group of class and . We will prove that .

An arbitrary element of can be written as

To show that , it is enough to show that , since order of is .

Now, for and , (9) becomes

which implies that

Also (9) for and gives that

That is, we have

Again (9), for , , gives that

Now (17) for gives that

Combining (61), (62), and (63), we get that

which together with (59) gives that .

Competing Interests

The author declares no competing interests.

References

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