Fixed Points of Automorphisms of Certain Non-Cyclic p-Groups and the Dihedral Group

: Let G = Z p ⊕ Z p 2 , where p is a prime number. Suppose that d is a divisor of the order of G . In this paper, we ﬁnd the number of automorphisms of G ﬁxing d elements of G and denote it by θ ( G , d ) . As a consequence, we prove a conjecture of Checco-Darling-Longﬁeld-Wisdom. We also ﬁnd the exact number of ﬁxed-point-free automorphisms of the group Z p a ⊕ Z p b , where a and b are positive integers with a < b . Finally, we compute θ ( D 2 q , d ) , where D 2 q is the dihedral group of order 2 q , q is an odd prime, and d ∈ { 1, q , 2 q } .


Introduction
Automorphisms of groups, algebras, Lie algebras and codes are of fundamental importance in many areas of mathematics and other disciplines; see, for more details, [3,6,7]. In [5], Farmakis and Moskowitz explain the importance of the fixed point theorems and their applications in many areas of mathematics including, but not limited to, analysis, algebraic groups, number theory, complex analysis and group theory. For example (cf. [5, theorem 5.4.1]), if α is an automorphism of a finite dimensional real or complex Lie algebra g having 0 as a fixed point, then the algebra g is nilpotent. Needless to say, nilpotent algebras appear naturally when one studies the structure of Lie algebras. On the other hand, the set of fixed points of an automorphism may contain useful information. For instance, in [8] Gersten proved a famous conjecture proposed by Scott that states that if φ is an automorphism of a finitely generated free group, then the set of fixed points of φ is finitely generated. Also, in areas such as image watermarking and live video streaming, toral automorphisms are used; see, for example, [7].
We now describe the main results of this paper. Let G be a finite group and let d be a divisor of the order of G. Then we define θ(G, d) as the number of automorphisms of G fixing d elements of G. In [2], the authors discuss the fixed points of automorphisms of some finite abelian groups. They study cyclic groups and elementary abelian groups. One of their main results is the derivation of formulas for θ(G, d), where d is a divisor of the order of G. In [2, p.66], the following conjecture is stated.
In this paper, we prove the above conjecture, see Theorem 3.10. In contrast to [9], where the explicit form of the automorphism group of Z p ⊕ Z p 3 is found, here we will make use of Theorem 3.1. This theorem, together with Lemmas 3.2, 3.3, 3.4, will allow us to find the explicit matrix form of the automorphism group of Z p ⊕ Z p 2 in a more concise way. In Section 4, we find the exact number of fixed-point-free automorphisms of the group Z p a ⊕ Z p b , where a and b are positive integers with a < b. Finally, in Section 5, we compute θ(D 2q , d); where D 2q is the dihedral group of order 2q, q is an odd prime, and d ∈ {1, q, 2q}. Throughout this paper, we denote the cardinality of a set X by |X|.

Preliminaries
Definition 2.1. Let G be a group. The set of all automorphisms of G under function composition forms a group, called the automorphism group of G and it is denoted by Aut G.
Definition 2.2. For a group G, the fixed point group of an automorphism f is defined as where Ω G is a map from Aut G to the collection of all subgroups of G. The map Ω G is known as the fixed point map.
For a non-empty subset X of G, we denote the set of automorphisms of G fixing at least all elements of X, as X Ω . That is Definition 2.6. Let n be a positive integer. Then the Euler's totient function ϕ(n) is defined as where (m, n) denotes the greatest common divisor of m and n.
Definition 2.7. Let n ≥ 2 be an integer and let Z n denote the group of integers modulo n. The group of units of Z n , which is denoted by Z * n , consists of all those congruence classes which are relatively prime to n; thus |Z * n | = ϕ(n).
Definition 2.8. Let G be a group and let φ : G → G be an automorphism of G. It is said that φ is a fixed-point-free automorphism if φ fixes only the identity element of G.
Definition 2.9. Let G be a group and let Aut G be its automorphism group. The holomorph of G is defined as the following semi-direct product Hol(G) = G ⋊ Aut G, where the product * is given by (g, α) * (h, β) = (gα(h), αβ).
Definition 2.10. For a positive integer n, the dihedral group D 2n is defined as the set of all rotations and reflections of a regular polygon with n sides.

Proof of Conjecture 1
In what follows, if S is a subset of a group G, then S denotes the subgroup generated by S.
By [11,Theorem 3.3] and [11,Theorem 4.2], the group Z p ⊕ Z p 2 has p 2 −1 p−1 = p + 1 subgroups of order p 2 , namely We first find the number of automorphisms of Z p ⊕ Z p 2 which fix at least p 2 elements. The strategy we will follow is to find the number of automorphisms that fix, at least, the above p + 1 subgroups of order p 2 and then substract the identity map.
The next theorem will play a crucial role in the proof of Conjecture 1. This theorem is proved in [1, Theorem 3.2]; however, for the reader's convenience, we include the statement.
where Z(G) denotes the center of the group G.
We will also make use of the following lemmas.

Lemma 3.2. There exists an isomorphism of abelian groups Hom
Proof. This is a standard fact; see, for example, [10, p.118]. Proof. This follows at once from the fundamental theorem of finitely generated abelian groups.
Lemma 3.4. The group Aut Z n is isomorphic to the group of units of Z n .
Proof. It suffices to consider the evaluation map T : Aut Z n → Z * n given by T (α) = α(1).
Using Lemmas 3.2, 3.3, 3.4 and Theorem 3.1, one obtains that the group Aut(Z p ⊕ Z p 2 ) may be realized as the following set of 2 × 2 matrices We now find the number of automorphisms of Z p ⊕ Z p 2 fixing p 2 elements. b) Now we treat the case Then we get the following pair of congruences Here we get pϕ(p) automorphisms and since there are p − 1 possible choices for k, then we obtain pϕ(p)(p − 1) = p(p − 1) 2 automorphisms.
c) Finally, we deal with J p = 1 0 , 0 p . We first find the form of those automorphisms that fix at least 1 0 ; then those that fix at least 0 p and take the intersection of these sets of matrices. Suppose first that φ is an automorphism such that φ 1 0 = 1 0 . This gives the following pair of congruences Hence a = 1 and c = 0. Thus φ has the form In this case, we get the following pair of congruences bp ≡ 0 (mod p) dp ≡ p (mod p 2 ) Therefore b is free and d ≡ 1 (mod p). In this case φ has the following form a b pc jp + 1 , where a ∈ Z * p , b, c, j ∈ Z p . Now define Consequently, we obtain p 2 automorphisms. Using a), b) and c) we obtain Proof. Immediate.

Proposition 3.8. The number of fixed-point-free automorphisms of
Proof. Let φ = a b pc d be an automorphism of Z p ⊕ Z p 2 . By Lemma 3.7, φ is a fixed-pointfree automorphism if and only if φ − id G is an automorphism of G. Note that and this will be an element of Aut(Z p ⊕ Z p 2 ) if and only if (a − 1, p) = 1 and (d − 1, p 2 ) = 1. Therefore, we need to count the number of elements a in Z * p such that (a − 1, p) = 1, and the number of d ∈ Z * p 2 such that (d − 1, p 2 ) = 1. Applying Lemma 3.6 we obtain p − 2 choices for a and p 2 − 2p choices for d. Since there are p choices for b and p choices for c, we obtain Proposition 3.9. We have that θ(Z p ⊕ Z p 2 , p) = p(2p 3 − 4p 2 + 1).
Proof. Using Remark 2 and Propositions 3.5, 3.8, we have Since the identity map is the unique automorphism fixing the entire group, then θ(Z p ⊕ Z p 2 , p 3 ) = 1. Combining Propositions 3.5, 3.8 and 3.9 we obtain the following Theorem 3.10. For the group G = Z p ⊕ Z p 2 , where p is any prime: This proves Conjecture 1.
Remark 3. In the case p = 2, we can also argue that θ(Z 2 ⊕ Z 4 , 1) = 0 using the following argument based on the characterization of Aut(Z 2 ⊕ Z 4 ). By [4, Lemma 11.1] we have that Aut(Z 2 ⊕ Z 4 ) ∼ = D 8 , the dihedral group of the square. Now, realize D 8 as the unitriangular matrix group of degree three over Z 2 (a.k.a the Heisenberg group modulo 2). It is easy to check that every matrix in this group has characteristic polynomial equal to (t − 1) 3 and thus 1 is always an eigenvalue. Therefore, no fixed-point-free automorphism of Z 2 ⊕ Z 4 exists.

Number of fixed-point-free automorphisms of Z p a ⊕ Z p b
Let a and b be positive integers with a < b. By [1, Theorem 3.2], Aut(Z p a ⊕ Z p b ) may be realized as the following set of 2 × 2 matrices: Proposition 4.1. Let a and b be positive integers with a < b. The number of fixed-point-free automorphisms of Z p a ⊕ Z p b is equal to: Proof. Let φ be an automorphism of G = Z p a ⊕ Z p b . Then, by Lemma 3.7, φ is a fixed-pointfree automorphism if and only if φ − id G is an automorphism. Therefore, it suffices to compute the cardinalities of the following sets Since there are p a possible choices for each β and c, then the total number of fixed-point-free automorphisms is equal to and the proof is now complete.
Remark 4. The above result implies that all the groups Z 2 a ⊕ Z 2 b , where 1 ≤ a < b, do not admit fixed-point-free automorphisms. This generalizes the result mentioned in Remark 3.

θ values for the dihedral group D 2p
The description of D 2n is obtained by using its generators: a rotation a of order n and a reflection b of order 2. Using rotations and reflections, we can write a presentation of D 2n D 2n = a, b : a n = b 2 = (ba) 2 = 1 = 1, a, a 2 , . . . , a n−1 , b, ab, a 2 b, . . . , a n−1 b .
Example 5.1. For n = 4, the dihedral group D 8 is the group of symmetries of the square. {1, a, a 2 , a 3 , b, ab, a 2 b, a 3 b} and its automorphism group is In the following table, the images of elements of D 8 under all automorphisms are given.
a a a a a 3 a 3 a 3 a 3 a 2 a 2 a 2 a 2 a 2 a 2 a 2 a 2 a 2 a 3 a 3 a 3 a 3 a 3 a a a a b b ab Finally, we end the paper with the following Question. Let p be a prime number and a 1 , . . . , a n be distinct positive integers. Is there an exact formula for θ n i=1 Z p a i , d ?

acknowledgments
We thank the anonymous referees for their comments and careful reading of the manuscript.