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Finite c-groups.

Abstract

We prove that in a finite group (p-group) the number of generators of the center of the group can be arbitrarily large and independent of the number of generators of the group.

AMS Subject Classification: 20xx.

Keywords: Group Theory and Generalizations.

1. Introduction

It seems counter-intuitive to think that the center of a group G, Z(G), may have an infinite number of generators, while G itself has only a fixed finite number of generators. In [Abels:1971], H. Abel gave an example of such a group. In this paper, for the case of finite groups, we extend [Blackburn:1972] and show the surprising result that in a finite group the number of generators of the center of the group can be arbitrarily large. That is, the number of generators of the center is not bounded by a function of the number of generators of the group.

Theorem 1.1. For all m; n [member of] N such that m [greater than or equal to] 2 and n [greater than or equal to] m, there exists a finite group G such that the minimal generating set of G has exactly m elements, and the minimal generating set of Z(G) has exactly n elements.

2. Notation and Definitions

Definition 2.1. [C-Group] If G is a group whose center has more generators than G, we say G is a C-Group.

Definition 2.2. [Commutator] Let G be a group and let x; y [member of] G. Then [x, y] = [xyx.sup.-1][y.sup.-1] is called the commutator of x and y. The subgroup of G generated by the set {[x, y]| x, : y [member of] G} is called the commutator subgroup of G and will be denoted G'.

3. Main Result

We begin by describing, briefly, the group that H. Abel gave, and then we present our result.

3.1. Infinite C-Groups

Here we briefly sketch a class of finitely generated groups whose centers are infinitely generated. One class of such groups is the set of 4 x 4 upper-triangular matrices over the ring [Q.sup.(p)] = {a/[p.sup.n]|a, n, p are integers, n [greater than or equal to] 0, p prime, and ([p.sup.n], a) = 1} [Abels:1971].

Define M([Q.sup.(p)], 4) to be all 4 x 4 matrices over [Q.sup.(p)] whose elements have the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [u.sub.i] denotes a positive unit. It is easy to check that this is a group under matrix multiplication.

It can be shown that M([Q.sup.(p)], 4) is finitely generated by the following elements: matrices where [u.sub.1] = p or 1/p while [u.sub.2] = 1 or the reverse, [u.sub.2] = p or 1/p while [u.sub.1] = 1 as well as matrices

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where exactly one [a.sub.i] = [+ or -]1 and all others are zero.

Now we examine the center of M([Q.sup.(p)], 4). Call it Z(M). We notice that for an element of M([Q.sup.(p)], 4) to be in Z(M), the element should be of the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Is Z(M) finitely generated? No! Assume it is. Then Z(M) must be generated by a finite number of central matrices. We call a set of such finite matrices A. Then there is an integer n such that [p.sup.n] is the maximum denominator among all the entries of the elements of the set A. Now we notice that the following matrix which is in Z(M) cannot be generated by elements of A:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

So we see that M([Q.sup.(p)], 4) is an example of a finitely generated group whose center isinfinitely generated.

3.2. Finite C-groups

First we prove the following Lemma.

Lemma 3.1. For all a, n, r, p [member of] [??] such that p is a prime and r [less than or equal to] [p.sup.n], we have ([ap.sup.n]/r).

[p.sup.r] [equivalent to] 0 mod [p.sup.n+1].

Proof. Write r! = [p.sup.k] x m, where k [member of] [??], k [greater than or equal to] 0, m [member of] [??] and p [??] m.

Now, as there are [r/p] numbers less than or equal to r which are divisible by p, [r/[p.sup.2]] which are divisible by [p.sup.2], and in general, [r/[p.sub.i]] which are divisible by [p.sup.i], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Now, let s = r - k. Clearly, s [member of] [??], where s [greater than or equal to] 1. Since a[p.sup.n] ... (a[p.sup.n] - r + 1)/m x [p.sup.k] = (apn r 2 N and (m; pn) = 1, then a(a[p.sup.n] - 1) ... (a[p.sup.n] - r + 1)/m = t [member of] [??]. It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

The main results of this paper, as stated in Theorem 1.1 in the introduction, are a corollary of the next theorem.

Theorem 3.2. The set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

together with the binary operation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is a C-group. It has two generators while its center has n + 2 generators.

Proof. In order to save space, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since [G.sub.p,n] is a semidirect product, it is a group.

Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. We will show that [g.sub.0] and [g.sub.1] are the generators of [G.sub.p,n]. Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and, using induction, we can show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

with [p.sup.k-1] in the kth position.

It can then be easily shown that any element in the group is expressible as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

and so [g.sub.0] and [g.sub.1] generate the group. Since the group is not abelian, it cannot be cyclic; therefore its minimal generating set must have exactly two elements.

Now, let x [member of] Z([G.sub.p,n]). Then x[g.sub.0] = [g.sub.0]x and x[g.sub.1] = [g.sub.1]x. We have x[g.sub.0] = [g.sub.0]x

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Also,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for some

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

By induction on h, we will show that for all i such that i [less than or equal to] min(h; n + 1), the ith entry of [M.sup.h] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is [p.sup.i] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and that the rest are 0:

For h = 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Let it hold for h. Then for h + 1 we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] = h

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] mod [p.sup.n+1] = 0 [left and right arrow] h mod [p.sup.n] = 0,

where p[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the second entry of [M.sup.h] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. By Lemma 3.1, the rest of the entries (after the first entry) will be equal to 0 mod [p.sup.n+1] if h mod [p.sup.n] = 0 as well.

So x must be of the form ([p.sup.n]h) mod [p.sup.n+1], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]), where h, [a.sub.1],..., [a.sub.n+1] [member of] [[??].sub.p]. Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

thus there exists an isomorphism

[phi]: Z([G.sub.p,n]) [right arrow] [n+2.summation over (i=1)] [[??].sub.p]

(the direct product of n + 2 copies of [[??].sub.p]), defined by

(([p.sup.n]h) mod [p.sup.n+1], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) [??] (h, [a.sub.1], [a.sub.2],..., [a.sub.n+1]).

Since the minimal generating set of [n+2.summation over (i=1)] [[??].sub.p] has exactly n + 2 elements, so does that of Z([G.sub.p,n]). As the minimal generating set of [G.sub.p,n] has 2 elements, and the minimal generating set of Z([G.sub.p,n]) has n + 2 elements, [G.sub.p,n] is a C-group.

This brings us to our main theorem

Theorem 3.3. For all m, n [member of] [??] such that m [greater than or equal to] 2 and n [greater than or equal to] m, there exists a finite group G such that the minimal generating set of G has exactly m elements, and the minimal generating set of Z(G) has exactly n elements.

Proof. The group [G.sub.p,n-m] x [m-2.summation over (i=1)] [[??].sub.p], for a prime p > 2, is an example of such a group.

References

[Abels:1971] Abels Herbert, 1971, An example of a finitely presented solvable group, London Mathematical Society lecture notes, Cambridge: Cambridge University Press, pp. 205-211.

[Blackburn:1972] Blackburn Norman, 1972, Some homology groups of wreathe products, Illinois J. Math. 16, pp. 116-129.

Kazem Mahdavi

University of Texas at Tyler

E-mail: kmahdavi@uttyler.edu

Brian Nguyen

University of California, San Diego

E-mail: brianjnguyen@gmail.com

Ramona R Ranalli

University of Texas at Tyler

E-mail: rranalli@uttyler.edu

James Sizemore

University of California, Los Angeles

E-mail: sizemmore@math.ucla.edu

Andrew Stout

University of California, San Diego

E-mail: astout@math.ucsd.edu

Colleen Swanson

Mt. Holyoke College

E-mail: cmswanso@mtholyohe.edu
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Author:Mahdavi, Kazem; Nguyen, Brian; Ranalli, Ramona R.; Sizemore, James; Stout, Andrew; Swanson, Colleen
Publication:Global Journal of Pure and Applied Mathematics
Geographic Code:1USA
Date:Apr 1, 2007
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