Muniya Sansanwal1⋆, Harsha Arora2 and Mahender Singh3
1Department of Mathematics, JJTU Jhunjhunu, Rajasthan, India
2Department of Mathematics, Govt. College for Women Hisar, Haryana, India
3Department of Mathematics, Om Sterling Global University, Hisar, Haryana, India
This paper contains the number of automorphisms of several non-Abelian groups of order p4, p-odd prime are computed and GAP (Group Algorithm Programming) software has been used for the verification of a number of automorphisms.
Keywords: -p-groups, automorphism, semi-direct product
MSC subject classification 2000: 20D45, 20D60
Let G be a p-group of order p4, p-odd prime and Aut(G) represent the group of all automorphisms of a given group G. There are many research papers in the literature related to the automorphisms of p-groups, for instance [1, 3, 4], etc. In [1], the automorphisms of groups of order p3 are computed along with the automorphisms of abelian groups of order p4. The present paper is an extension of the research work in [1]. In this paper, we will compute the automorphisms of groups of order p4. William Burnside [2] classified all groups having order p4. By using these classifications we derive the automorphisms of some groups having order p4. We divide the derived results into two parts. In these two parts, one part is dedicated to the categorization of all p−groups of order p4 and the other part is dedicated to investigating the number of automorphisms of several non-Abelian groups with order p4.
According to the categorization derived by Burnside [2], if p is an odd prime, then there will be 15 groups of order p4. Five of these are abelian and the rest are non-abelian, as given below.
Abelian Groups:
Non-Abelian Groups:
The automorphisms of five abelian groups, namely G1, G2, G3, G4, and G5, of order p4 are already computed in the paper [1]. In this paper, we compute the automorphisms of several non-abelian p−groups with order p4, namely G9, G10, and G11 denoted above.
We can deduce relations by using generator relations from the structure description of G9 and elementary calculations:
Now, we shall investigate the automorphism group
Let be defined by:
As ψ is automorphism, ψ should preserve the relation x2x1x3 = x3x1, that is
As n ≢ 0(mod p), because if n ≡ 0(mod p) then order of (ψ(x2)) ≠ p.
Also, x4x3 = x3x4 then ψ(x4)ψ(x3) = ψ(x3)ψ(x4)
Further, x1x4 = x4x1 implies ψ(x1)ψ(x4) = ψ(x4)ψ(x1)
Now, ys − uw ≡ 0(mod p) and yi − kw ≡ 0(mod p) are two homogeneous equations with determinant iu − sk ≠ 0. Hence, these equations have only trivial solutions. ⇒ y = w ≡ 0(mod p).
Next, we check the kernel of ψ. Let be any element of
so that it is mapped to identity.
Equations (20.1) and (20.3) are the system of two homogeneous equations with iu − sk ≠ 0 determinant, which gives only a trivial solution. Hence, h1 = h3 = 0. From (20.4), we get
As
we get
Further, from (20.2) we get nh2 ≡ 0(mod p), but n ≢ 0(mod p) ⇒ h2 = 0. So, if and ψ(h) = 1, it must be that h = 1.
Hence, the kernel is trivial and ψ is an automorphism with the constraints on the parameters already deduced. Now it is very easy to calculate the order of the automorphism group. We have p choices for all j, l, t, v, x, and p − 1 choices for z and i, u, s, and k will in some sense be equivalent to a matrix in GL(Fp), which gives (p2 −p)(p2 − 1) choices for the elements. Therefore, we have that is really defined as:
By using our relations from the structure of G10 and elementary calculations, we can deduce some useful relations:
We begin our study of automorphism group by using the above relations.
Let be defined by:
Since ψ is an automorphism, ψ should preserve the relation x4x3 = x2x3x4, hence
using (20.7) and (20.8) in (20.5) and (20.6), we get
Also,
Using Equations (20.13) and (20.14) in (20.11) and (20.12), we get
Since j, k, l ≡ 0(mod p), therefore i ≢ 0(mod p) because if i ≡ 0(mod p), then Ord(x1) ≠ p. Further, from (20.15), i = zn ≢ 0(mod p), we have z ≢ 0(mod p) and n ≢ 0(mod p).
Next, we have x2x3 = x3x2
Now, n ≢ 0(mod p) gives v ≡ 0(mod p)
From (20.10), we can see zu ≡ n + vy(mod p), therefore zu ≡ n(mod p) or n ≡ zu(mod p).
Also, from (20.15) i ≡ zn(mod p), we get i ≡ zzu(mod p). Thus, we have
Further, as n ≡ zu(mod p) and n ≢ 0(mod p), therefore u ≢ 0 (mod p). Substituting the value of v in (20.9), we get
Now, we shall find the constraints on the parameters for which ψ has a trivial kernel. Let
From (20.20), we have h4 ≡ 0(mod p) for z ≢ 0(mod p) and from (20.19), we get uh3 ≡ 0(mod p). Since u ≢ 0(mod p), therefore h3 = 0. Further, from (20.18), we have nh2 ≡ 0(mod p). As n ≢ 0(mod p), we have h2 = 0. From (20.17), ih1 ≡ 0(mod p), but i ≢ 0(mod p), thus h1 = 0.
So, if and ψ(h) = 1, it must be that h = 1, hence the kernel is trivial and ψ is an automorphism with the constraints we have already deduced. Thus, for ψ to be an automorphism there are p choices for all s, t, w, x, y, and p − 1 choices for both u and z. Therefore, we have that
is really defined as:
We can deduce some useful relations by using the generator relations of G11 and elementary calculations:
With these relations, we begin our study of the automorphisms of
Let be defined by:
Now, ord(x1) = ord(ψ(x1)) = p2.
Therefore,
Also, the order of is p, so the order of ψ(x2) is
Order of x3 is also p, so Order of ψ(x3) is p
As ψ is an automorphism, ψ should preserve the relation thus we get
Next, we find the constraints for parameters using the kernel. Let be any element of
that it is mapped to identity.
From (20.21), we get
Next, from (20.22) we have
Also, from (20.23) we obtain sz ≡ 0(mod p), but p f s, therefore z ≡ 0(mod p).
So, if and ψ(h) = 1, it must be that h = 1, hence the kernel is trivial and ψ is an automorphism with the restrictions we have already deduced on the parameters. Now, we have here p2 − p choices for i and p choices for all l, q, j, k, n, and p − 1 choices for s. Therefore, we have that
is really defined as: