Institute of Mathematics and Natural Sciences, Tallinn University, Narva mnt. 25,10120 Tallinn, Estonia
Received Date: January 20, 2008; Revised Date: April 13, 2008
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Let n,m be integers such that n ¸ 3, m > 0 and Ck a cyclic group of order k. All groups which can be presented as a semidirect product (C2n+m × C2n) h C2 are described.
All non-Abelian groups of order < 32 are described in  (Table 1 at the end of the book). M. Jr. Hall and J. K. Senior  have given a fully description of all groups of order There exist exactly 51 non-isomorphic groups of order 32. Some of them can be presented as a semidirect product and some of them as a semidirect product As a generalization of the first case, in  all groups of the form are described. It turned out that there exist only 17 non-isomorphic groups of this form (for a fixed
n). In this paper we generalize the second case. Namely, we shall describe all finite 2-groups
which can be presented in the form Clearly, each such group G is given by three generators a, b, c and by the defining relations
for some and the ring of residue classes modulo 2k).
The aim of this paper is to prove
Theorem 1.1. For fixed and the number of groups which can be given by relations
All possible values of (p, q, r, s) are given in Propositions 3.1, 3.2, 3.3 if m < n, in 3.1, 3.2 if m = n and in 3.4, 3.5 if m > n.
Let be a group given by (1.1). An element c induces an inner automorphism of order two (the case = 1 is also included) of group
Therefore, we have to find all automorphisms of of order two. The map induces an endomorphism of group if and only if (mod ). This endomorphism
is an automorphism, if and only if (mod 2). This map is an automorphism of order two if and only if (p, q, r, s) satisfy the system
Our purpose is to solve system (2.1). Note that the two first subsystems of (2.1) imply the
following system modulo 2n
Let be a solution of system (2.2), where and denotes the set of all invertible elements of Then p and r can be replaced in (2.1) by
Now it is easy to see that system (2.1) is equivalent to the system
where and and (2.4)
Remark, that means the representative of residue class; moreover, we always can choose
Because the length of the paper is limited, for most of statements we give only idea of proof.
Assume that Then and system (2.3) takes the form
Proposition 3.1. Assume that and q is odd. Then the solutions (p, q, r, s) of (2.1) are of the form where and where There are exactly solutions of this form if exactly solutions if m = 2 and exactly solutions if m = 1.
Proof. The condition of the proposition, conditions (2.4) and are satisfied
for solution of (2.2) from the set .While we have Where and if if Since and (mod 2m), the second congruence of (3.1) holds for every From
the first congruence of (3.1) we get the value for y. Now let us find the number of solutions of the system (2.1). We have choices for number choices for odd number choices for
number x. For i0 we have z = 4 choices if choices if m = 2 and z = 1 choice if m = 1. This implies that for the number i we have choices and the number of solutions of the system is equal to the number of triples and
Proposition 3.2. Assume that q is even and Then the
solutions of (2.1) are:
where and and in the case if then in the case then where and There are exactly solutions of this form if and solutions if m = n.
There are exactly 32 solutions of this form.
Proof. To prove the proposition, by  we must consider the following sets of solutions of (2.2):
where Solving system (3.1) for each solution of (2.2) from given sets we get from the
second congruence in (3.1) the condition for y and from the first congruence in (3.1) the values for x. The solutions of system (2.2) belonging to set vgive us solution 1) of system
(2.1). The solutions of system (2.2) belonging to sets give solution 2) of
Proposition 3.3. Assume that and are both nonzero even numbers, is odd and Then system (2.1) have solutions only if and these solutions are where and if then if there are solutions of this form. If there are solutions.
Proof. Let us now consider the set The solutions of system (2.2) from this set have the form where and (3.2)
The condition holds only if The second congruence of (3.1), i.e
holds in the case if k = 0 for every and in the case if k = 1 it holds for every Since the first congruence of (3.1), i.e
Since this congruence holds if and only if
The last condition is stronger than (3.2) and implies where is the inverse of the odd number u by modulo Since for v we have values by modulo in the form
It follows from (3.3), that in the case m = 1 we have and in the case m > 1 we have
Calculating the number of all obtained solutions, we get the second statement of proposition
The case m > n
The condition implies g = 0 and i.e y is even, where System (2.3) has now
Lemma 3.1. The solution of the congruence
Proof. The solutions of are i.e. if f=1 Then
Lemma 3.2. The solution of the congruence
where and and
Proof. Denote then Using (2.1)–(2.10)
in , we get that the solution of the last congruence is
Now let us find x. Since it follows that Analogously .if then and then
Denote by x1 solutions from Lemma 3.1 and by x2, z2 solutions from Lemma 3.2.
Proposition 3.4. Assume that m > n and the number q is odd Then the solutions of (2.1) are: then the where and There are solutions of these forms.
Proof. Consider the solutions of system (2.2) belonging to the set M3. The second congruence
of (3.4) holds for every To solve the first congruence of (3.4), consider two cases for z: In the first case using Lemma 3.2, we
get solution 1) and in the second case, using Lemma 3.1, we get solution 2).
Proposition 3.5. Assume that m > n, and both numbers q and g are
even. Then (2.1) have solutions only in case f = i = ±1 and these solutions are:
where there are solutions of these forms.
Proof. Consider solutions of system (2.2) belonging to the sets Solving system (3.4) and using lemmas 3.1 and 3.2, we get from the set solutions 1), 2), 3), 4) and from sets solution 5). Calculating the number of all obtained
solutions, we get the second statement of the proposition.
Research was supported by the Estonian Science Foundation Research Grant 5900, 2004-2007.