On Equalities of Central Automorphism Group with Various Automorphism Groups

Abstract

In this chapter, we intend to present a survey on the equalities of central automorphisms of a group and pose some problems and further research directions.

The second author is thankful to DST for the funding support under the file no MTR/2022/000331

6.1 Introduction

Throughout the chapter, p denotes a prime number. For group G, we denote by G^′, Z(G), cl(G), d(G), Φ(G), and Aut(G), respectively, the commutator subgroup, the center, the nilpotency class, the rank, the Frattini subgroup, and the automorphism group of G. An automorphism σ of group G is called central if σ commutes with every automorphism in Inn(G), the group of inner automorphisms of G, (equivalently, if g⁻¹σ(g) lies in the center Z(G) of G, for all g in G.)

The central automorphisms of G fix the commutator subgroup of G elementwise and form a normal subgroup of the full automorphism group Aut(G); we denote this subgroup by Aut_z(G) in this paper. For groups G having Aut(G) abelian, it is necessarily the case that Aut_z(G) = Aut(G). The non-abelian groups G with Aut(G) abelian are called as Miller groups (see [19]). However, several people constructed various groups G for which Aut(G) is non-abelian and Aut_z(G) = Aut(G) (see [7, 11, 15, 18]). In 2001, Curran and McCaughan [6] considered the case where the central automorphisms are just the inner automorphisms of G, that is, Aut_z(G) = Inn(G); one can also see [4, 23]. Continuing in this direction, in 2004, Curran [8], for group G, derived the equality ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}},$; the same is derived in [1, 12, 22]. Let ${Aut_z^{z}}(G)$ be the set of all central automorphisms of a group G which fixes the center Z(G) of G elementwise. In 2007, Attar [2] characterized finite $\operatorname {\mathrm {p-groups}}$ for which ${Aut_z^{z}}(G)={Inn}(G)$ holds. In 2009, Yadav [25] characterized $\operatorname {\mathrm {p-groups}}$ of nilpotency class 2 for which ${Aut_z}(G)={Aut_z^{z}}(G)$ (for the same equality, also see [14]).

An $\operatorname {\mathrm {automorphism}}$ ϕ of a group G is called class preserving if ϕ(x) is conjugate to x for all x ∈ G. The set Aut_c(G) of all class-preserving automorphisms of G forms a normal subgroup of Aut(G) and contains Inn(G). In 2013, Yadav [26] characterized finite $\operatorname {\mathrm {p-groups}}$, and Kalra and Gumber [16] characterized all finite $\operatorname {\mathrm {p-groups}}$ of order ≤ p⁶ (for any prime p) and ≤ p⁵ (for odd prime p) for which the set of all central automorphisms is equal to the set of all class-preserving automorphisms, that is, Aut_z(G) = Aut_c(G); the same equality is derived in [10].

An automorphism σ of a group G is called IA − automorphism if it induces the identity automorphism on the abelian quotient G∕G^′. Let IA_z(G) be the group of those IA automorphisms which fix the center of G elementwise. In 2014, Rai [21] characterized finite $\operatorname {\mathrm {p-groups}}$ in which Aut_z(G) = IA_z(G) if and only if γ₂(G) = Z(G). In 2016, Kalra and Gumber [17], characterized finite non-abelian $\operatorname {\mathrm {p-groups}}$ G for which Aut_z(G) = IA_z(G) if and only if G^′ = Z(G).

Hegarty [13] defined the notions of absolute center and autocommutator of a group G (analogous to Z(G) and G^∗ as follows:

$$\displaystyle \begin{aligned} L(G)=\{ g\in G|\operatorname{\mathrm{\alpha}}(g)=g\,\forall \operatorname{\mathrm{\alpha}}\in {Aut}(G) \} \end{aligned}$$

$$\displaystyle \begin{aligned} G^*=\langle g^{-1}\operatorname{\mathrm{\alpha}}(g)| g\in G,\, \operatorname{\mathrm{\alpha}}\in{Aut}(G)\rangle\end{aligned}$$

These are clearly characteristic subgroups of G. Also, Z(G) ⊃ L(G) and G^′⊂ G^∗. Hegarty [13] also defined absolute central $\operatorname {\mathrm {automorphism}}$ of G as follows: an $\operatorname {\mathrm {automorphism}}$ γ of a group G is called an absolute central $\operatorname {\mathrm {automorphism}}$ if it induces identity $\operatorname {\mathrm {automorphism}}$ on G∕L(G). The set Aut_l(G) of all absolute central automorphisms of G forms a normal subgroup of Aut(G); it is also a subgroup of Aut_z(G). Let ${Aut_{l}^z}(G)$ denote the group of absolute central automorphisms of G which fix Z(G) elementwise.

In 2020, Singh and Gumber [24] gave a necessary and sufficient condition on finite $\operatorname {\mathrm {p-group}}$ G for which Aut_z(G) = Aut_l(G) and also for which ${Aut_z}(G)={Aut_{l}^z}(G)$.

6.2 Equalities of Central Automorphisms

6.2.1 Equalities with Group of All Automorphisms

Definition 6.1

Following Earnley, a non-abelian group with abelian automorphism group is called Miller group.

If Aut(G) is abelian, then it is clear that all the automorphisms are central, i.e., Aut_z(G) = Aut(G). The obvious examples of groups with abelian automorphism group are the cyclic groups. There are non-abelian groups with abelian automorphism groups; these are called Miller groups (seeEarnley [9]). Several researchers constructed various examples of groups for which Aut_z(G) = Aut(G), even if Aut(G) is non-abelian. Curran, in 1982, found first such example. He constructed a group of order 2⁷ for which Aut_z(G) = Aut(G) and Aut(G) is non-abelian.

Theorem 6.1 ([7], Proposition, p. 394)

There exists a non-abelian group G of order 2⁷ which has a non-abelian automorphism group of order 2¹² in which every automorphism is central, that is, Aut_z(G) = Aut(G).

Example of such group is given below:

Let M be the Miller group of order 2⁶, and let

$$\displaystyle \begin{aligned} G=M\times Z_2=\langle a,b,c,d\,|\,a^8=b^4=c^2=d^2=1,\, a^b=a^5,\,b^c=b^{-1},\,\end{aligned}$$

$$\displaystyle \begin{aligned}{}[a,c]=[a,d]=[b,d]=[c,d]=1 \rangle\end{aligned}$$

This result of Curran leads the motivation to $\operatorname {\mathrm {p-groups}}$ for p an odd prime in which Aut_z(G) = Aut(G) and Aut(G) is non-abelian. In 1984, Malone proved the following result:

Theorem 6.2 ([18], Proposition, p. 36)

For each odd prime p, there exists a non-abelian p-group with a non-abelian automorphism group in which each automorphism is central, that is, Aut_z(G) = Aut(G).

For each odd prime p, we consider the group

$$\displaystyle \begin{aligned} F=\langle a_1,\,a_2,\,a_3,\,a_4,\,|\,(a_i,\,a_j,\,a_k)=1\, and \,a_i^{p^2}=1 \,\,for\end{aligned}$$

$$\displaystyle \begin{aligned} 1\leq i,j,k\leq 4;\, (a_1,a_2)=a_{1}^{p};\,(a_1,a_3)=a_{3}^{p};\,(a_1,a_4)=a_{4}^{p}\end{aligned}$$

$$\displaystyle \begin{aligned} (a_2,a_3)=a_{2}^{p};\,(a_2,a_4)=1\,;(a_3,a_4)=a_{3}^{p}\rangle\end{aligned}$$

Aut(F) is abelian group. We set B = 〈 b| b^p = 1 〉. Group G = F × B is non-abelian group which has Aut(G) non-abelian in which each automorphism is central.

Curran in [7] and Malone in [18] derived the examples of groups with direct factors for which Aut_z(G) = Aut(G) and Aut(G) is non-abelian. The question was left if there is a group G with no direct factors for which Aut_z(G) = Aut(G) and Aut(G) is non-abelian. Continuing in this direction, in 1986, Glasby produced an infinite family of 2-groups having no direct factors and which have a non-abelian automorphism group in which all automorphisms are central.

Definition 6.2

Define G_n to be the group generated by x₁, …, x_n $x_{i}^{2^{i}}=1 \,\,\,\, (1 \leq i \leq n)$ $[ x_{i},x_{i+1}]=x_{i+1}^{2^{i}},\,\,\,\,(1 \leq i < n)$. [x_i, x_j] = 1, (1 < i + 1 < j ≤ n)

Theorem 6.3 ([11], Theorem, p. 234)

For n ≥ 3, G_n has no direct factors, and Aut(G_n) is non-abelian of order 2^p(n), where p(n) = (n − 1)(2n² − n = 6∕6)(n ≥ 4), in which every $\operatorname {\mathrm {automorphism}}$ is central.

In 2012, Jain and Yadav [15] constructed the following family of groups G_n with no direct factor, for which Aut_z(G_n) = Aut(G_n).

Definition 6.3

Let n be a natural number greater than 2 and p an odd prime. Define G_n to be the group generated by x₁, x₂, x₃, x₄

$$\displaystyle \begin{aligned} x_{1}^{p^{n}}=x_{2}^{p^{3}}=x_{3}^{P^{2}}=x_{4}^{p^{2}}=1,\end{aligned}$$

$$\displaystyle \begin{aligned}{}[x_{1},x_{2}]=x_{2}^{p^{2}},\,\,\,\,[x_{1},x_{3}]=x_{3}^{p}\end{aligned}$$

$$\displaystyle \begin{aligned}{}[x_{1},x_{4}]=x_{4}^{p},\,\,\,\,[x_{2},x_{3}]=x_{1}^{p^{n-1}}\end{aligned}$$

$$\displaystyle \begin{aligned}{}[x_{2},x_{4}]=x_{2}^{p^{2}},\,\,\,\, [x_3,x_4]=x_4^p .\end{aligned}$$

This group G is a regular p-group of nilpotency class 2 having order pⁿ⁺⁷ and exponent pⁿ. Further, Z(G) =  Φ(G) and therefore G is purely non-abelian.

Theorem 6.4 ([15], Theorem A, p. 228)

Let m = n + 7 and p be an odd prime, where n is a positive integer greater than or equal to 3. Then there exists a group G of order p^m, exponent pⁿ, and with no nontrivial abelian direct factor such that Aut_z(G) = Aut(G) is non-abelian.

6.2.2 Equalities with Group of Inn(G) and $\operatorname {\mathrm {Z(Inn(G))}}$

In 2001, Curran and McCaughan [6] characterized finite $\operatorname {\mathrm {p-groups}}$ in which central automorphisms are precisely the inner automorphisms.

Theorem 6.5 ([6], Theorem, p. 2081)

If G is a finite p-group, then Aut_z(G) = Inn(G) if and only if G^′ = Z(G) and Z(G) is cyclic.

Definition 6.4

A group G, whose only element of finite order is the identity, is called torsion-free group.

Definition 6.5

A non-abelian group G is purely non-abelian if G has no nontrivial abelian direct factor.

In 2016, Azhdari characterized all finitely generated groups G for which the equality Aut_z(G) = Inn(G) holds. He proved the following:

Theorem 6.6 ([4], Theorem 2, p. 4134)

Let G be a finitely generated group. Then Aut_z(G) = Inn(G) if and only if one of the following assertion holds:

• G is purely non-abelian and Z(G) = G^′ is cyclic.

• G≅C₂ × N where N is purely non-abelian with |Z(N)| odd and Z(N) = N^′ is cyclic (or Z(G) = C₂ × G^′ is cyclic).

• G is torsion-free with Z(G) = G^′ is cyclic and det(M_G) = 1 where M_G is skew-symmetric matrix corresponding to G.

In 2018, Sharma et al. [23] verified the equality Aut_z(G) = Inn(G) for the finite p-groups of order up to p⁷ as follows:

Theorem 6.7 ([23], Theorem 2.1, p. 3)

There is no $\operatorname {\mathrm {p-group}}$ G of order up to p⁶ satisfying Aut_z(G) = Inn(G).

Theorem 6.8 ([23], Theorem 2.2, p. 3)

A $\operatorname {\mathrm {p-group}}$ G of order p⁷ satisfies Aut_z(G) = Inn(G) if and only if $Z(G)\cong C_p^2$, |G^′| = p⁴ and cl(G) = 4.

In 2004, Curran [8] considered the case where the central automorphism group is as small as possible. Clearly, $\operatorname {\mathrm {Z(Inn(G))}} \leq {Aut_z}(G)$, for any group G. When G is arbitrary, Aut_z(G) and $\operatorname {\mathrm {Z(Inn(G))}}$ may coincide because both these subgroups of Aut(G) can be trivial. However, the situation becomes interesting if G is a p-group, since both subgroups are nontrivial.

Theorem 6.9 ([8], Theorem 1.1, p. 223)

Let G be a finite non-abelian p-group. If ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$, then Z(G) ≤ G^′, and furthermore, ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if $\operatorname {\mathrm {Hom}}(G/G,Z(G)) \approx Z(G/Z(G))$.

In 2013, Sharma and Gumber [22] characterized p-groups of order ≤ p⁵(for any prime p) and of order p⁶(for p odd), for which ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$.

Theorem 6.10 ([22], Theorem 3.2, p. 3)

Let G be p-group of order p⁵ and cl(G) = 3. Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if d(G) = 2 and Z(G)≅C_p.

Theorem 6.11 ([22], Theorem 3.3, p. 3)

Let G be a p-group of order p⁶, for an odd prime p, and cl(G) = 3 0r4. Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if d(G) = 2 and Z(G)≅C_p.

Continuing the study of Curran [8] of minimum order of Aut_z(G), Gumber and Kalra [12] obtained the following:

Let $G/G^{\prime } \cong C_{p^{r_1}}\times \dots C_{p^{r_{n}}}$ (r₁ ≥⋯ ≥ r_n ≥ 1) and $Z_{2}(G)/Z(G) \cong C_{p^{s_1}}\times \dots C_{p^{s_{m}}}$ (s₁ ≥⋯ ≥ s_m ≥ 1).

Theorem 6.12 ([12], Theorem 2.1, p. 1803)

Let G be a finite p-group with $Z(G)\cong C_{p^{b_1}}$. Then ${Aut_z} (G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if either G∕G^′≅Z₂(G)∕Z(G) or d(G) = d(Z₂(G)∕Z(G)), s_i = b₁ for 1 ≤ i ≤ c, and s_i = r_i for c + 1 ≤ i ≤ n, where c, 1 ≤ c ≤ n is the largest such that r_c ≥ b₁.

Definition 6.6

The coclass of a finite $\operatorname {\mathrm {p-group}}$ G of order pⁿ is n − c, where c is the class of the group.

Corollary 6.1 ([12], Corollary 2.2, p. 1804)

Let G be a finite p-group of coclass 2. Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if Z(G)≅C_p and d(G) = d(Z₂(G)∕Z(G)) = 2.

Corollary 6.2 ([12], Corollary 2.3, p. 1804)

Let G be a finite p-group of coclass 3. Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if Z(G)≅C_p and d(G) = d(Z₂(G)∕Z(G)) = 2, 3 or $Z(G)\cong C_{p^2}$ and Z₂(G)∕Z(G)≅G∕G^′.

Corollary 6.3 ([12], Corollary 2.4, p. 1804)

Let G be a finite p-group of coclass 4. Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if one of the following conditions holds:

• (a) Z(G)≅C_p and d(G) = d(Z₂(G)∕Z(G)) = 2, 3, 4.

• (b) $Z(G)\cong C_{p^2}$ and either Z₂(G)∕Z(G)≅G∕G^′ or $Z_2(G)/Z(G)\cong C_{p^2}\times C_p$ and $G/G^{\prime } \cong C_{p^3}\times C_p$ or $Z_2(G)/Z(G)\cong C_{p^2}\times C_p$ and $G/G^{\prime } \cong C_{p^4}\times C_p$.

• (c) $Z(G)\cong C_{p^3}$ and Z₂(G)∕Z(G)≅G∕G^′.

Gumber and Kalra also generalized the results of Sharma and Gumber [22] as follows:

Theorem 6.13 ([12], Theorem 3.1, p. 1804)

Let G be p-group of order = p⁵ and cl(G) = 3. Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if Z(G)≅C_p and d(G) = d(Z₂(G)∕Z(G)) = 2.

Theorem 6.14 ([12], Theorem 3.2, p. 1805)

Let G be a finite $\operatorname {\mathrm {p-group}}$ such that cl(G) = 3 or 4. Then, ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if Z(G)≅C_p and d(G) = d(Z₂(G)∕Z(G)) = 2.

Also, Gumber and Kalra obtained the result for |G| = p⁷ as in [22]; it was up to p⁶.

Theorem 6.15 ([12], Theorem 3.3, p. 1805)

Let G be a p-group of order p ⁷ . Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if one of the following holds:

• cl(G) = 3, Z(G) ≃ C_p and rank(G) = rank(Z₂(G)∕Z(G)) = 2, 3, 4.

• cl(G) = 4 and either Z(G) ≃ C_p and rank(G) = rank(Z₂(G)∕Z(G)) = 2, 3 or Z(G) is cyclic group of order p² and Z₂(G)∕Z(G) ≃ G∕G^′.

• cl(G) = 5, Z(G) ≃ C_p and rank(G) = rank(Z₂(G)∕Z(G)) = 2.

Let G be a non-abelian p-group G. Let $G/G^{\prime }\cong C_{p^{c_{1}}}\times C_{p^{c_{2}}}\times \dots \times C_{p^{c_{r}}}$ (c₁ ≥⋯ ≥ c_r ≥ 1) and $Z_{2}G/Z(G)\cong C_{p^{d_{1}}}\times C_{p^{d_{2}}}\times \dots \times C_{p^{d_{s}}}$ (d₁ ≥ d₂ ≥…d_s ≥ 1), where $C_{p^{a_i}}$ is a cyclic group of order $p^{a_i}.$

In 2020, Attar [1] characterized the finite p-groups in some special cases, including p-groups G with C_G(Z( Φ(G)) ≠  Φ(G), $\operatorname {\mathrm {p-groups}}$ with an abelian maximal subgroup, metacyclic $\operatorname {\mathrm {p-groups}}$ with p ≥ 2, $\operatorname {\mathrm {p-groups}}$ of order pⁿ and exponent pⁿ⁻², and Camina $\operatorname {\mathrm {p-groups}}$, for which Aut_z(G) is of minimal order, as follows:

Theorem 6.16 ([1], Theorem 3.1, p. 4)

Let G be a finite $\operatorname {\mathrm {p-group}}$ such that C_G(Z( Φ(G)) ≠  Φ(G). Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if Z(G) is cyclic and one of the following is true:

• G∕G^′≅Z₂(G)∕Z(G).

• r = s, d_i = h for 1 ≤ i ≤ t, d_i = c_i for t + 1 ≤ i ≤ r, where p^h = exp(Z(G)) and t is the largest integer between 1 and s such that c_t > h.

Corollary 6.4 ([1], Corollary 3.2, p. 5)

Let G be a non-abelian finite p-group with an abelian maximal subgroup. Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if G^′ = Z(G) and Z(G) is cyclic.

Theorem 6.17 ([1], Theorem 3.3, p. 6)

Let G be a non-abelian metacyclic finite p-group with p > 2. Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if Z(G) ≤ G^′.

Corollary 6.5 ([1], Corollary 3.4, p. 6)

The finite non-abelian $\operatorname {\mathrm {p-groups}}$ G of order p ⁿ and exponent p ⁿ⁻¹ for which ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ are of the following isomorphism types:

• (1) $M(p^3)=\langle \operatorname {\mathrm {\alpha }}, \operatorname {\mathrm {\beta }} | \operatorname {\mathrm {\alpha }}^{p^2}= \operatorname {\mathrm {\beta }}^p=1,\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^{1+p}\rangle (p>2)$.

• (2) $D_8=\langle \operatorname {\mathrm {\alpha }}, \operatorname {\mathrm {\beta }} | \operatorname {\mathrm {\alpha }}^4= \operatorname {\mathrm {\beta }}^2=1,\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^{-1}\rangle$.

• (3) $Q_8=\langle \operatorname {\mathrm {\alpha }}, \operatorname {\mathrm {\beta }} | \operatorname {\mathrm {\alpha }}^4=1,\, \operatorname {\mathrm {\beta }}^2= \operatorname {\mathrm {\alpha }}^2,\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^{-1}\rangle$.

Corollary 6.6 ([1], Corollary 3.5, p. 7)

Let p be an odd prime. Then finite non-abelian $\operatorname {\mathrm {p-groups}}$ of order p ⁿ and exponent p ⁿ⁻² for which ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ are one of the following isomorphism types:

• (1) $G=\langle \operatorname {\mathrm {\alpha }},\, \operatorname {\mathrm {\beta }},\, {\gamma }| \operatorname {\mathrm {\alpha }}^p= \operatorname {\mathrm {\beta }}^p={\gamma }^p=1,\, \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\beta }} \operatorname {\mathrm {\alpha }},\, {\gamma }^{-1} \operatorname {\mathrm {\alpha }}{\gamma }= \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }},\, \operatorname {\mathrm {\beta }}{\gamma }={\gamma } \operatorname {\mathrm {\beta }} \rangle$.

• (2) $G=\langle \operatorname {\mathrm {\alpha }},\, \operatorname {\mathrm {\beta }} | \operatorname {\mathrm {\alpha }}^{p^3}= \operatorname {\mathrm {\beta }}^{p^2}=1,\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^{1+p}\rangle$.

• (3) $G=\langle \operatorname {\mathrm {\alpha }},\, \operatorname {\mathrm {\beta }} | \operatorname {\mathrm {\alpha }}^{p^4}= \operatorname {\mathrm {\beta }}^{p^2}=1,\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^{1+{p^2}}\rangle$.

Corollary 6.7 ([1], Corollary 3.6, p. 8)

The finite non-abelian 2-groups G of order 2ⁿ and exponent 2ⁿ⁻² for which ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ are one of the following:

• (1) $G=\langle \operatorname {\mathrm {\alpha }},\, \operatorname {\mathrm {\beta }},\, {\gamma }| \operatorname {\mathrm {\alpha }}^8= \operatorname {\mathrm {\beta }}^2={\gamma }^2=1,\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^5,\, {\gamma }^{-1} \operatorname {\mathrm {\alpha }}{\gamma }= \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }},\, \operatorname {\mathrm {\beta }}{\gamma }={\gamma } \operatorname {\mathrm {\beta }} \rangle$.

• (2) $G=\langle \operatorname {\mathrm {\alpha }},\, \operatorname {\mathrm {\beta }},\, {\gamma }| \operatorname {\mathrm {\alpha }}^{2^{n-2}}=1,\, \operatorname {\mathrm {\beta }}^2=1,\,{\gamma }^2= \operatorname {\mathrm {\beta }},\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^{1+{2^{n-3}}},\, {\gamma }^{-1} \operatorname {\mathrm {\alpha }}{\gamma }= \operatorname {\mathrm {\alpha }}^{-1} \operatorname {\mathrm {\beta }}, \rangle$.

• (3) $G=\langle \operatorname {\mathrm {\alpha }}, \, \operatorname {\mathrm {\beta }} | \operatorname {\mathrm {\alpha }}^{16}= \operatorname {\mathrm {\beta }}^4=1,\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^5\rangle$.

• (4) $G=\langle \operatorname {\mathrm {\alpha }}, \, \operatorname {\mathrm {\beta }} | \operatorname {\mathrm {\alpha }}^{2^{n-2}}=1,\, \operatorname {\mathrm {\beta }}^4=1,\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^{-1+2^{n-4}}\rangle$, where n ≥ 6.

• (5) $G=\langle \operatorname {\mathrm {\alpha }},\, \operatorname {\mathrm {\beta }},\, {\gamma }| \operatorname {\mathrm {\alpha }}^{2^{n-2}}=1,\, \operatorname {\mathrm {\beta }}^2=1,\,{\gamma }^2=1,\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^{1+{2^{n-3}}},\, {\gamma }^{-1} \operatorname {\mathrm {\alpha }}{\gamma }= \operatorname {\mathrm {\alpha }}^{-1+2^{n-4}} \operatorname {\mathrm {\beta }}, \, \operatorname {\mathrm {\beta }}{\gamma }={\gamma } \operatorname {\mathrm {\beta }} \rangle$, where n ≥ 6.

• (6) $G=\langle \operatorname {\mathrm {\alpha }},\, \operatorname {\mathrm {\beta }},\, {\gamma }| \operatorname {\mathrm {\alpha }}^{2^{n-2}}=1,\, \operatorname {\mathrm {\beta }}^2=1,\,{\gamma }^2= \operatorname {\mathrm {\alpha }}^{2^{n-3}},\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^{1+{2^{n-3}}},\, {\gamma }^{-1} \operatorname {\mathrm {\alpha }}{\gamma }= \operatorname {\mathrm {\alpha }}^{-1+2^{n-4}} \operatorname {\mathrm {\beta }}, \, \operatorname {\mathrm {\beta }}{\gamma }={\gamma } \operatorname {\mathrm {\beta }} \rangle$, where n ≥ 6.

• (7) $G=\langle \operatorname {\mathrm {\alpha }}, \operatorname {\mathrm {\beta }}, {\gamma }| \operatorname {\mathrm {\alpha }}^8=1,\, \operatorname {\mathrm {\beta }}^2=1,\, {\gamma }^2= \operatorname {\mathrm {\alpha }}^4,\, \operatorname {\mathrm {\beta }}^{-1} \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\alpha }}^5,\, {\gamma }^{-1} \operatorname {\mathrm {\alpha }}{\gamma }= \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }},\, \operatorname {\mathrm {\beta }}{\gamma }={\gamma } \operatorname {\mathrm {\beta }} \rangle$.

A pair (G, N) is called Camina pair if 1 < N < G is normal subgroup of G and for every element g ∈ G∕N, the element g is conjugate to all gN.

Theorem 6.18 ([1], Theorem 3.7, p. 12)

Let G be a non-abelian finite $\operatorname {\mathrm {p-group}}$ such that (G, Z(G)) is a Camina pair. Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if Z(G)≅C_p and G∕G^′≅Z₂(G)∕Z(G).

Theorem 6.19 ([1], Corollary 3.8, p. 12)

Let G be a finite non-abelian Camina $\operatorname {\mathrm {p-group}}$. Then ${Aut_z}(G)= \operatorname {\mathrm {Z(Inn(G))}}$ if and only if G^′ = Z(G) and Z(G) is cyclic.

6.2.3 Equalities with Class-Preserving Automorphisms

For a finite $\operatorname {\mathrm {p-group}}$ G, the subgroup Ω_m(G) is defined as $\langle x\in G|\,\, x^{p^m}=1\rangle$, and ℧_m(G) is defined as $\langle x^{p^m}|\,x\in G \rangle$. For a finite $\operatorname {\mathrm {p-group}}$ G with cl(G) = 2 , G∕Z(G) is abelian. Consider the following cyclic decomposition of G∕Z(G) : 

$$\displaystyle \begin{aligned} G/Z(G)\cong C_{p^{e_1}}\times\ldots\times C_{p^{e_k}}\,\,\, (e_1\geq e_2\geq\dots\geq e_k\geq1).\end{aligned}$$

In 2013, Yadav (see [26]) and Kalra and Gumber (see [16]) characterized $\operatorname {\mathrm {p-groups}}$ of class 2 with Aut_z(G) = Aut_c(G) as follows:

Theorem 6.20 ([26], Theorem A, p. 2)

Let G be a finite $\operatorname {\mathrm {p-group}}$ of class 2. Then Aut_z(G) = Aut_c(G) if and only if G^′ = Z(G) and $|{Aut_{c}(G)}|=\Pi _{i=1}^{d}|\Omega _{m_i}(G^{\prime })|$

Theorem 6.21 ([26], Theorem B, p. 2)

Let G be a finite $\operatorname {\mathrm {p-group}}$ and cl(G) = 2 with Aut_z(G) = Aut_c(G) and then rank of G is even.

Theorem 6.22 ([16], Theorem 3.1, p. 3)

Let G be a finite $\operatorname {\mathrm {p-group}}$. Then Aut_z(G) = Aut_c(G) if and only if Aut_c(G)≅Hom(G∕Z(G), G^′) and G^′ = Z(G).

Theorem 6.23 ([16], Theorem 3.3, p. 4)

Let G be a finite non-abelian $\operatorname {\mathrm {p-group}}$ such that the center of the group is elementary abelian. Then Aut_z(G) = Aut_c(G) if and only if G is a Camina $\operatorname {\mathrm {p-group}}$ and cl(G) = 2.

Theorem 6.24 ([16], Theorem 3.4, p. 4)

Let G be a finite non-abelian $\operatorname {\mathrm {p-group}}$ such that Z(G) is cyclic. Then Aut_z(G) = Aut_c(G) if and only if Z(G) = G^′.

Definition 6.7

A finite $\operatorname {\mathrm {p-group}}$ G of class 2 is said to have property (∗) if for some π $\mho _{m_{i}^{\pi }}(\Omega _{n_i}(Z(G))\leq [x,G]$ for all x ∈ G∕Z(G) and i ∈{1, …, k}.

In 2015, Ghoraishi found a necessary and sufficient condition for a finite $\operatorname {\mathrm {p-group}}$ G to satisfy Aut_z(G) = Aut_c(G), as follows:

Theorem 6.25

Let G be a finite $\operatorname {\mathrm {p-group}}$. Then Aut_z(G) = Aut_c(G) if and only if Z(G) = G^′ and G has property (∗).

6.2.4 Equalities with Absolute Central and IA Automorphisms

Definition 6.8

A finite non-Abelian group G is said to be purely non-Abelian if it has no nontrivial Abelian direct factor.

Let $C_{{Aut}(G)}({Aut_{l}}(G))= \{ \operatorname {\mathrm {\alpha }}\in {Aut}(G)\,| \operatorname {\mathrm {\alpha }} \operatorname {\mathrm {\beta }}= \operatorname {\mathrm {\beta }} \operatorname {\mathrm {\alpha }},\,\forall \, \operatorname {\mathrm {\beta }}\in {Aut_{l}}(G)\}$ denote the centralizer of Aut_l(G) in Aut(G). In [20], Moghaddam and Safa defined $E(G)=[G,C_{{Aut}(G)}({Aut_{l}}(G))]=\langle g^{-1} \operatorname {\mathrm {\alpha }}(g)\,|g\in G,\, \operatorname {\mathrm {\alpha }}\in C_{Aut(G)}({Aut_{l}}(G))\rangle .$ One can easily see that E(G) is a characteristic subgroup of G containing the derived group G^′ = [G, Inn(G)], and each absolute central $\operatorname {\mathrm {automorphism}}$ of G fixes E(G) elementwise [20, Theorem C].

Let

$$\displaystyle \begin{aligned} {G/E(G) \cong C_{p^{e_{1}}}\times C_{p^{e_{2}}}\times \dots \times C_{p^{e_{k}}}}, \,\,\,(e_{1}\geq \dots e_{k}\geq 1)\end{aligned}$$

$$\displaystyle \begin{aligned} {G/G^{\prime} \cong C_{p^{f_{1}}}\times C_{p^{f_{2}}}\times \dots \times C_{p^{f_{l}}}},\,\,\,(f_{1}\geq \dots f_{l}\geq 1)\end{aligned}$$

$$\displaystyle \begin{aligned} L{(G )\cong C_{p^{g_{1}}}\times C_{p^{g_{2}}}\times \dots \times C_{p^{g_{m}}}},\,\,\,(g_{1}\geq \dots g_{m}\geq 1)\end{aligned}$$

$$\displaystyle \begin{aligned} Z(G)\cong C_{p^{h_{1}}}\times C_{p^{h_{2}}}\times \dots\times C_{p^{h_{n}}}\,\,\,(h_{1}\geq \dots h_{n}\geq 1).\end{aligned}$$

Since G∕E(G) is a quotient group of G∕G^′, it follows that k ≤ l and e_i ≤ f_i for all 1 ≤ i ≤ k.

In the same year, M. Singh and D. Gumber [24] obtained the equalities of Aut_z(G) with Aut_l(G), the group of absolute central automorphisms, and ${Aut_{l}^z}(G)$, the group of absolute central automorphisms that fix the center elementwise, as follows:

Theorem 6.26 ([24], Theorem 1, p. 864)

Let G be a finite non-Abelian $\operatorname {\mathrm {p-group}}$. Then ${Aut_z}(G)={Aut_{l}^z}(G)$ if and only if either L(G) = Z(G) or Z(G) ≤ Φ(G), G^′ = E(G), m = n, and e₁ ≤ g_t, where t is the largest integer between 1 and m such that g_t < h_t.

Theorem 6.27 ([24], Theorem 2, p. 865)

Let G be a finite non-abelian $\operatorname {\mathrm {p-group}}$ such that L(G) < Z(G). Then ${Aut_z}(G)={Aut_{l}^z}(G)$ if and only if Z(G) ≤ Φ(G), G^′ = E(G)Z(G), m = n, e₁ ≤ g_t, where t is the largest integer between 1 and m such that g_t < h_t.

In 2014, Rai [21] characterized finite $\operatorname {\mathrm {p-groups}}$ for which Aut_z(G) = IA_z(G), where IA_z(G) denote the group of those IA automorphisms which fix the center elementwise, as follows:

Theorem 6.28 ([21], Theorem B(1), p. 170 )

Let G be a finite p-group. Then Aut_z(G) = IA_z(G) if and only if G^′ = Z(G).

Let X and Y  be the two finite abelian $\operatorname {\mathrm {p-groups}}$, and let $X\cong C_{p^{a_{1}}}\times C_{p^{a_{2}}}\times \dots \times C_{p^{a_{i}}}$ and $Y\cong C_{p^{b_{1}}}\times C_{p^{b_{2}}}\times \dots \times C_{p^{b_{j}}}$ be the cyclic decomposition of X and Y , where a_t ≥ a_t+1 and b_s ≥ b_s+1 are positive integers. If either X is proper subgroup or proper quotient group of Y  and d(X) = d(Y ), then there certainly exists r, 1 ≤ r ≤ i such that a_r < b_r, a_k = b_k for r + 1 < k < i. For this unique fixed r, let var(X, Y ) = p^r. In other words, var(X, Y ) denotes the order of the last cyclic factor of X whose order is less than that of corresponding cyclic factor of Y.

In 2016, Kalra and Gumber obtained Aut_z(G) = IA_z(G) for finite non-abelian $\operatorname {\mathrm {p-groups}}$ as follows:

Theorem 6.29 ([17], Theorem 2.12, p. 5)

Let G be a finite non-abelian $\operatorname {\mathrm {p-group}}$. Then Aut_z(G) = IA_z(G) if and only if either G^′ = Z(G) or G^′ < Z(G), d(G^′) = d(Z(G)) and exp(G∕G^′) ≤ var(G^′, Z(G)).

6.2.5 Equalities with Central Automorphisms Fixing the Center Elementwise

In 2007, Attar [2] characterized groups in which the central automorphisms fixing the center elementwise are precisely inner automorphisms, as follows:

Theorem 6.30 ([2], Theorem, p. 297)

If G is a $\operatorname {\mathrm {p-group}}$ of finite order, then ${Aut_z^{z}}(G)={Inn}(G)$ if and only if G is abelian or nilpotency class of G is 2 and Z(G) is cyclic.

Let G be a finite p-group of class 2. Then G∕Z(G) and G^′ have equal exponent p^C(say). Let

$$\displaystyle \begin{aligned} G/Z(G)\cong C_{p^{c_{1}}}\times C_{p^{c_{2}}}\times \dots\times C_{p^{c_{m}}}\,\,\,(c_1\geq \dots \geq c_m\geq 1)\end{aligned}$$

where $C_{p^{c_{i}}}$ is a cyclic group of order $p^{c_{i}}$, 1 ≤ i ≤ r. Let k be the largest integer between 1 and r such that c₁ = c₂ = c_k = e. Note that k ≥ 2. “Let M be the subgroup of G containing Z(G) such that

$$\displaystyle \begin{aligned} \bar{M}=M/Z(G)= C_{p^{c_{1}}}\times C_{p^{c_{2}}}\times \dots\times C_{p^{c_{k}}}.\text{''}\end{aligned}$$

Let

$$\displaystyle \begin{aligned} G/G^{\prime}\cong C_{p^{d_{1}}}\times C_{p^{d_{2}}}\times \dots\times C_{p^{d_{n}}}\,\,\,d_{1}\geq d_{2}\geq \dots d_{s}\geq 1\end{aligned}$$

be a cyclic decomposition of G∕G^′ such that $\bar {M}$ is isomorphic to a subgroup of

$$\displaystyle \begin{aligned} \bar{N}=N/G^{\prime}:= C_{p^{d_{1}}}\times C_{p^{d_{2}}}\times \dots\times C_{p^{d_{k}}}.\end{aligned}$$

In 2009, using the above terminology, Yadav proved the following:

Theorem 6.31 ([25], Theorem, p. 4326)

Let G be a finite p-group of class 2. Then ${Aut_z}(G)={Aut_z^{z}}(G)$ if and only if m = n, $G/Z(G)/\bar {M}\cong (G/G^{\prime })/\bar {N}$, and exp(Z(G)) = exp(G^′).

In 2011, Azhdari and Akhavan-Malayeri [5] generalized the result of Attar in [2] for the finitely generated groups of nilpotency class 2. They got the following:

Theorem 6.32 ([5], Theorm 0.1, p. 1284)

Let G be a finitely generated of cl(G) = 2. Then ${Aut_z^{z}}(G)={Inn}(G)$ if and only if Z(G)≅C_p or $Z(G) \cong C_{n}\times \mathbb {Z}^{s}$ where exp(G∕Z(G))∕n and s is torsion-free rank of Z(G).

Theorem 6.33 ([5], Corollary 0.2)

Let G be a finitely generated group of class 2, which is not torsion-free. Then ${Aut_z^{z}}(G)={Inn}(G)$ if and only if cl(G) = 2 and Z(G) is cyclic or $Z(G) \cong C_{n}\times \mathbb {Z}^{s}$ with exp(G∕Z(G)) divides n and s is torsion-free rank of Z(G).

Theorem 6.34 ([5], Corollary 0.3)

Let G be a finitely generated of cl(G) = 2. G^′ is torsion-free, and ${Aut_z^{z}}(G)={Inn}(G)$ if and only if Z(G) is infinite cyclic.

In the same year, Jafari also found a necessary and sufficient condition on a finite $\operatorname {\mathrm {p-group}}$ G such that ${Aut_z}(G)={Aut_z^{z}}(G)$, as follows:

Theorem 6.35

Let G be a finite $\operatorname {\mathrm {p-group}}$ . Then ${Aut_z}(G)={Aut_z^{z}}(G)$ if and only if $Z(G)G{'}\subseteq G^{p^n}G{'}$ , where $\exp (Z(G))=p^n.$

Let G be a non-abelian finite p-group. Let

$$\displaystyle \begin{aligned} G/G^{\prime}= C_{p^{c_{1}}}\times C_{p^{c_{2}}}\times \dots\times C_{p^{c_{r}}}\,\,\, (c_{1}\geq \dots c_{r}\geq 1).\end{aligned}$$

$$\displaystyle \begin{aligned} G/G^{\prime} Z(G)\cong C_{p^{d_{1}}}\times C_{p^{d_{2}}}\times \dots\times C_{p^{d_{s}}}\,\,\,(d_{1}\geq \dots d_{s}\geq 1).\end{aligned}$$

and $Z(G)\cong C_{p^{e_{1}}}\times C_{p^{e_{2}}}\times \dots \times C_{p^{e_{t}}}\,\,\,(e_{1}\geq \dots e_{t}\geq 1).$

since G∕G^′Z(G) is a quotient of G∕G^′.

In 2012, Attar [3] gave a necessary and sufficient condition on finite $\operatorname {\mathrm {p-group}}$ G such that Aut_z(G) to be ${Aut_z^{z}}(G)$, as follows:

Theorem 6.36 ([3], Theorem A, p. 1097)

Let G be a non-abelian finite $\operatorname {\mathrm {p-group}}$. Then ${Aut_z}(G)={Aut_z^{z}}(G)$ if and only if Z(G) ≤ G^′ or Z(G) ≤ Φ(G), r = s, and c₁ ≤ b_m where m is the largest integer between 1 and r such that a_m > b_m.

Theorem 6.37 ([3], Corollary 2.1, p. 1098)

Let G be a non-abelian finite $\operatorname {\mathrm {p-group}}$ such that exponent of Z(G) is p. Then ${Aut_z}(G)={Aut_z^{z}}(G)$ if and only if Z(G) ≤ Φ(G).