On equality of derival and inner automorphisms of some p-groups

For a group G, D(G) denotes the group of all derival automorphisms of G. For a finite nilpotent group of class 2, it is shown that D(G) ≅ Hom(G∕ 2 (G), 2 (G)). We prove that if G is a nilpotent group of class ≥ 3 such that Z(G) ⊆ γ 2 (G) and D(G∕Z(G)) = Inn(G∕Z(G)), then D(G) = Inn(G) if and only if Autcent(G) = Z(Inn(G)). Finally, for an odd prime p, we classify all p-groups of order p, 1 ≤ n ≤ 5, for which D(G) = Inn(G). Subjects: Advanced Mathematics; Algebra; Group Theory; Mathematics & Statistics; Pure Mathematics; Science


Introduction
Let G be a group. Notations used are standard, however for the sake of completeness, by e we denote the identity element of G. For x, y ∈ G, x y denotes the conjugate element y −1 xy and [x, y] = x −1 y −1 xy is the commutator of x and y. x G denotes the conjugacy class of x in G. The subgroup generated by the set of all commutators of G is called derived group of G and it is denoted by [G, G] or G

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Groups are the most basic algebraic structure which is the building block of modern algebra. Groups handle many practical problems like symmetries of objects and various problems of combinatorics. Automorphism group is one of the most fascinating object associated with a group. In recent past, many researchers proposed different automorphism groups and their equality has been established with each other. This article is one in this sequence. The notion of derival automorphism group is introduced and its equality with the groups of inner automorphisms and classpreserving automorphisms is discussed. Results obtained are very fundamental in nature and will be quite helpful for researchers working in group theory.
group of all class-preserving automorphisms is denoted by Aut c (G). The group of all inner automorphisms of G is denoted by Inn(G) and it is a normal subgroup of Aut c (G). Out c (G) denotes the quotient group Aut c (G)∕Inn(G). Let N be a characteristic subgroup of G. Then each ∈ Aut(G), induces an automorphism :G∕N → G∕N given by (xN) = (x)N. Thus the map :Aut(G) → Aut(G∕N) given by ( ) = is a homomorphism of groups. The kernel of this homomorphism is precisely those automorphisms of G which are identity on G/N. If we take N = G � , then Ker is the group of all those automorphisms of G which are identity on G∕G � . This group is abbreviated as D(G) and elements of this group are called derival automorphisms of G (ChiŞ, 2002). An automorphism of G is called central if it is identity on G/Z(G). The set of all central automorphisms of G is a normal subgroup of Aut(G) and it is denoted by Autcent(G). It has been shown by Sah (1968) that Autcent(G) = C Aut(G) (Inn(G)). In the recent past, interest of many mathematicians turned on the equalities of various automorphism groups viz. equalities of Autcent(G) and Inn(G), Autcent(G) and Z(Inn(G)) and Autcent(G) and Aut c (G) etc. Curran and McCaughan (2001) showed that if G is a finite p-group, then Autcent(G) = Inn(G) if and only if G � = Z(G) and Z(G) is cyclic. Further extending this work, Curran (2004) observed that Autcent(G) is minimum possible when Autcent(G) = Z(Inn(G)) and he found that if Autcent(G) = Z(Inn(G)), then Z(G) ⊆ G � and Z(Inn(G)) must be cyclic. Gumber and Sharma (2011) proved that if G is a nilpotent group of class 2, then Autcent(G) = Z(Inn(G)) if and only if G � = Z(G) and Z(G) is cyclic. For a finite p-group, Jafari (2011) find out the necessary and sufficient condition when every central automorphism fix Z(G) element-wise. Jain (2012) studied those finite p-groups for which Aut(G) = Autcent(G). Kalra and Gumber (2013) characterize all finite p-groups of order p n , 1 ≤ n ≤ 7, such that Aut c (G) = Autcent(G). Further Yadav (2013) proved that if G is a finite pgroup such that Aut c (G) = Autcent(G), then G has even number of elements in any minimal generating set for G. Ghoraishi (2015) find out the necessary and sufficient condition for equality of class preserving and central automorphism of a finite group. Vermani (2000, 2001) show that if G is a group of order p n , 1 ≤ n ≤ 4, then Aut c (G) = Inn(G). In another note Yadav (2008) studied class-preserving automorphisms of group of order p 5 , p an odd prime and proved that Aut c (G) = Inn(G) for all groups G of order p 5 except two isoclinism families. On the classification for group of order p 6 given by James (1980), Narain and Karan (2014) studied those groups of order p 6 for which Aut c (G) = Inn(G).
If is either an inner automorphism or a class-preserving automorphism of G, then one should . This shows that This motivate us to study those p-groups for which D(G) coincides with Aut c (G) or Inn(G). This paper is an attempt to study some p-groups for which D(G) = Aut c (G). One quite natural situation when We characterize all finite p-groups of class 2 for which D(G) = Inn(G). For an odd prime p, we also classify those groups of order p 5 for which D(G) = Inn(G).

Preliminaries and definitions
This section deals with some of the basic definitions and results which are used further.
Definition 1 For a group G, the sequence {Z i } i≥0 of subgroups of G defined as follows is called the upper central series of G; its i-th term is called the i-th center of G. Here Z 1 = Z(G), the center of G. A group G is said to be nilpotent if Z m = G, for some positive integer m. The smallest integer c such that Z c (G) = G, is called the nilpotency class of G.
Definition 3 A group G is called nilpotent group of class 2 if G has a lower central series of the form The quaternion group Q 8 is an example of a nilpotent group of class 2. In fact every non-abelian group of order, p 3 is a nilpotent group of class 2.

Definition 4 A group is called a Camina group if and only if
Every abelian group is a Camina group trivially. Q 8 =< a, b|a 4 = 1, b 2 = a 2 , b −1 ab = a −1 >, the quaternion group is a non-abelian p-group of order 8 with |Z(Q 8 )|=p. Note that derived group of Q 8 coincides with the center and it is a non-abelian Camina p-group (for proof see Lemma 2.1).
Definition 5 Let G be a group in which each element is of finite order, then the exponent of G is the least common multiple of orders of all elements and it is denoted by exp(G) .
Following results are important for further study.

Lemma 2.1 Let G be a p-group of class 2. Then for each
Proof Let G be a nilpotent group of class 2. Then G

Nilpotent groups of class 2
If G is an abelian group, then derived group G ′ for such a group is trivial and hence Inn(G), Aut c (G) and D(G) contain merely the identity automorphism. Since abelian groups are precisely nilpotent group of class 1, this motivate us to study D(G) for nilpotent groups of class 2.
Proof Let G be a nilpotent group of class 2. Then for each f ∈ D(G), the map f :G ⟶ 2 (G)(⊆ Z(G)) defined by f (x) = x −1 f (x), is a homomorphism of groups. Since f sends elements of 2 (G) to 1, it induces homomorphism f :G∕ 2 (G) ⟶ 2 (G) given by f (x 2 (G)) = x −1 f (x). Thus we have the map :D(G) ⟶ Hom(G∕ 2 (G), 2 (G)) given by (f ) = f . Let f , g ∈ D(G) and x ∈ G − 2 (G). Then there exists a ∈ 2 (G) such that g(x) = xa. Since f sends elements of 2 (G) to 1, This shows that is a homomorphism of groups.
Define a map :G ⟶ G by (x) = xf (x 2 (G)). It is easy to see that is an endomorphism of G. Now x −1 (x) = f (x 2 (G)) ∈ 2 (G). Let x ∈ ker( ). Then there are only two possibilities that either . Thus x ∈ ker if and only if x ∈ 2 (G). But then 1 = (x) = xf (x 2 (G)) = xf ( 2 (G)) = x. This shows that ker = {1} and hence is a monomorphism. Since G is a finite group, is an automorphism of G. Thus ∈ D(G).

Lemma 3.2 Let G be a p-group of class 2. Then, the order of each non-trivial element xZ(G) in G/Z(G) is equal to the exponent of the subgroup [x, G].
Proof It is well known that in a finite p-group of class 2, exp( 2 (G))= exp (G/Z(G)).
Let G be a finite p-group of order p n . Let {x 1 , x 2 ⋯ x r } be a minimal generating set for G. If | 2 (G)| = p m , then by Burnside basis theorem r ≤ n − m. The following remarkable theorem (Theorem 5.1, Yadav, 2007) is quite useful in our context. (2) If {x 1 , x 2 ⋯ x r } be a minimal generating setfor G, then (2) Let {x 1 , x 2 ⋯ x r } be a minimal generating set for G. Then (b) Since for a finite p-group of class 2, exp ( 2 (G)) = exp (G / Z(G)), from part (a), it follows that for

Theorem 3.5 Let G be a finite p-group of class 2. Then Aut c (G) = D(G) if and only if G is a Camina p-group.
Proof If G is a Camina p-group, then G Conversely suppose that D(G) = Aut c (G). Let |G| = p n and |G G], it follows that |Aut c (G)| = |G � | r = (p m ) r = (p m ) n−m . Thus by theorem 3.3, G is a Camina special p-group. ✷ Remark Since Camina special p-group is a particular kind of Camina groups, the above result holds good for Camina special p-groups.

Groups of order p 4
In next two sections, we study groups of order p 4 and p 5 (p is an odd prime), on the basis of the classification given by James (1980). This classification is given in terms of isoclinism families. We start with the following definition of isoclinism of groups, given by Hall (1940).
Let X be a finite group and X = X∕Z(X). Then commutation in X gives a well-defined map X :X × X ↦ 2 (X) such that X (xZ(X), yZ(X)) = [x, y] for (x, y) ∈ X × X. ], for all , ∈ G, where � Z(H) = ( (Z(G)) and � Z(H) = ( Z(G)). The resulting pair ( , ) is called an isoclinism of G onto H. Clearly isomorphic groups are isoclinic but isoclinic groups need not be isomorphic. For example, Q 8 and D 8 are isoclinic groups which are not isomorphic. If G and H are isoclinic groups, then i (G) ≅ i (H) and i (G) ≅ i (H), whereas it is not necessary that Z(G) ≅ Z(H) (Hall, 1940). But one may observe that if G and H are finite isoclinc groups of equal order, then |Z(G)| = |Z(H)|. Since our further classification depends on the size of 2 (G) and that of Z(G), it is sufficient to calculate |D(G)| for only one member from each isoclinism family. :G∕Z(G) → H∕Z(H), : 2 (G) → 2 (H),