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Action of Aut(G) on the set of maximal subgroups of p-groups

Publié en ligne: 05 Sep 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 25 Jan 2022
Accepté: 15 May 2022
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Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Introduction

Redei [1] gave the definition of a minimal non-abelian group G, whose subgroups are abelian except G itself, and categorised them into three categories: quaternion group, metacyclic groups and non-metacyclic groups. Hermann [2] gave the composition of groups, whose all maximal subgroups are pairwise isomorphic, with remainder classes numbering 2. Mann [3] gave the relationship between the number of remainder classes and nilpotent classes of G, whose maximal subgroups are isomorphic to each other. The complete classification of maximal automorphism p-groups with nilpotent class 3 can be seen in a previous paper [4].

It is an important and difficult subject in group theory to study the effect of the structure of group G on Aut(G). For some abelian groups, their automorphism groups are non-abelian. Therefore, the nature of the group does not necessarily hold for its automorphism group. It is not easy to find the composition of Aut(G), even if it is not easy to find the order of Aut(G). There are some results on the order of Aut(G) in other previous works [5,6,7,8].

In this thesis, we study the action of Aut(G) on the set of all maximal subgroups of some p-groups.

Preliminaries

In this section, we give some known definitions and lemmas, which will be frequently used.

Definition 1

[9]. Let Ω = {α, β, γ,⋯} and SΩ denote a symmetric group on Ω. A homomorphism φ from group G to SΩ is called an action of G on Ω. In other words, each element xG corresponds to a one-to-one transformation φ(x) of Ω, where φ(x) : ααx, satisfying (αa)b = αab, a, bG, α ∈ Ω, α ∈ Ω. If Ker φ = G, then we say that G acts on Ω trivially.

Definition 2

[10, 11]. For α ∈ Ω, αG = {αa |aG} is called the orbit of G containing α.

Definition 3

[12,13,14]. If G has only one orbit on Ω, i.e. Ω itself, we say that G acts on Ω transitively.

Definition 4

[12, 15, 16]. The number of elements in an orbit is called the length of the orbit.

Lemma 1

[17,18,19,20]. Suppose that K is a subgroup of group G, τAut(G). Then, (1) Kτ is a subgroup of G and K is isomorphic to Kτ; (2) τ−1Aut(G).

Lemma 2

[1]. Let G be a minimal non-abelian p-group. Then, G is one of the following groups:

Quaternion group <a,b|a4=1,b2=a2,ab=a1>. < a,{\kern 1pt} b|{a^4}{\kern 1pt} = {\kern 1pt} 1,{\kern 1pt} {b^2}{\kern 1pt} = {\kern 1pt} {a^2},{\kern 1pt} {a^b}{\kern 1pt} = {\kern 1pt} {a^{ - 1}} > .

Metacyclic group Mp(n,m)=<a,b|apn=bpm=1,ab=a1+pn1>,withn2,m1 {M_p}(n,{\kern 1pt} m){\kern 1pt} = {\kern 1pt} < a,{\kern 1pt} b|{a^{{p^n}}}{\kern 1pt} = {\kern 1pt} {b^{{p^m}}}{\kern 1pt} = {\kern 1pt} 1,{\kern 1pt} {a^b}{\kern 1pt} = {\kern 1pt} {a^{1 + {p^{n - 1}}}} > ,\;with\;n \ge 2,{\kern 1pt} m \ge 1

Non-metacyclic group Mp(n,m,1)=<a,b|apn=bpm=cp=1,[a,b]=c,[a,c]=[b,c]=1>,wherem+n3ifp=2. {M_p}(n,{\kern 1pt} m,{\kern 1pt} 1){\kern 1pt} = {\kern 1pt} < a,{\kern 1pt} b|{a^{{p^n}}} = {b^{{p^m}}} = {c^p} = 1,{\kern 1pt} [a,{\kern 1pt} b] = c,{\kern 1pt} [a,{\kern 1pt} c] = [b,{\kern 1pt} c] = 1 > ,\;where\;m + n \ge 3\;if\;p = 2.

Lemma 3

[21, 22]. A finite p-group G is minimal non-abelian if and only if Φ(G) = Z(G) and d(G) = 2.

Lemma 4

[23]. Suppose that G is a finite p-group, G =< x, y >, τAut(G), where τ : x → xiyj, y → xrys. Then, G =< xτ, yτ >⇔ isjr ≡ kp + l, where kZ, l = 1, 2,⋯ , p − 1.

Lemma 5

[23]. Let G =< x, y|xpn = ypm = 1, xy = x1+pn−1 >, with n ≥ 2, m ≥ 1. Then,

if p = 2, n = m = 2, we have Aut(G) =< τ1, τ2, τ3, τ4 >, where τ1:xx2n1,yy;τ2:xy2x,yy;τ3:xx,yyx;τ4:xx,yy2m1. {\tau _1}:x \to {x^{{2^n} - 1}},{\kern 1pt} y \to y;{\tau _2}:x \to {y^2}x,{\kern 1pt} y \to y;{\tau _3}:x \to x,{\kern 1pt} y \to yx;{\tau _4}:x \to x,{\kern 1pt} y \to {y^{{2^m} - 1}}.

if p = 2, m ≥ 3, 3 ≤ n ≤ m, we have Aut(G) =< τ1, τ2, τ3, τ4, τ5, τ6 >, where τ1:xx5,yy;τ2:xx2n1,yy;τ3:xy2mn+1x,yy;τ4:xx,yyx;τ5:xx,yy5;τ6:xx,yy2m1 \matrix{ {{\tau _1}:x \to {x^5},{\kern 1pt} y \to y;{\kern 1pt} {\tau _2}:x \to {x^{{2^n} - 1}},{\kern 1pt} y \to y;{\kern 1pt} {\tau _3}:x \to {y^{{2^{m - n + 1}}}}x,{\kern 1pt} y \to y;} \hfill \cr {{\tau _4}:x \to x,{\kern 1pt} y \to yx;{\kern 1pt} {\tau _5}:x \to x,{\kern 1pt} y \to {y^5};{\kern 1pt} {\tau _6}:x \to x,{\kern 1pt} y \to {y^{{2^m} - 1}}} \hfill \cr }

if p > 2, mn, we have Aut(G) =< τ1, τ2, τ3, τ4 > ã < τ5 >, where τ1:xxp+1,yy;τ2:xypx,yy;τ3:xx,yyx;τ4:xx,yy1+p; {\tau _1}:x \to {x^{p + 1}},{\kern 1pt} y \to y;{\kern 1pt} {\tau _2}:x \to {y^p}x,{\kern 1pt} y \to y;{\kern 1pt} {\tau _3}:x \to x,{\kern 1pt} y \to yx;{\kern 1pt} {\tau _4}:x \to x,{\kern 1pt} y \to {y^{1 + p}};

τ5 : x → xs, y → y, s = tpn−1, t is the primitive root of modulo pn.

Results and discussion

In this article, we study the finite p-group G; Ω = {K | K is any maximal subgroup of G}, and we use P to represent the action of Aut(G) on Ω.

Action
Theorem 3.1.1

Suppose that G is a p-group, τAut(G), assume P(τ):KKτ,KΩ. P(\tau ):K \to {K^\tau },{\kern 1pt} K \in \Omega . Then, P is an action of Aut(G) on Ω.

Proof

Since K is maximal, we obtain that Kτ is a subgroup of G, under isomorphism τ.

Next, we can prove that Kτ is maximal. Assume that Kτ is a proper set of M and M is a subgroup of G. By Lemma 1, τ−1 ∈ Aut(G), we obtain that K is a proper set of Mτ−1. Because Mτ−1 is a subgroup of G and K is maximal, we have Mτ−1 = G. Then, M = G, under isomorphism τ−1. Thus, Kτ is a maximal in G. So, P(τ) is a map from Ω to itself.

If K1 and K2 are subgroups of G, where K1K2, then K1τK2τ K_1^\tau \ne K_2^\tau , under isomorphism τ.

So, P(τ) is an injection from Ω to itself. Moreover, P(τ) is a surjection from Ω to itself because Ω is a finite set. So, P(τ) is a one-to-one transformation of Ω. That is, P(τ) ∈ SΩ. Thus, P is a map from Aut(G) to SΩ.

In addition, ∀K ∈ Ω, Kτσ = (Kτ)σ , τ, σ ∈ Aut(G). So, P is a homomorphism from Aut(G) to SΩ. This completes the proof of the theorem.

We notice that K is isomorphic to Kτ, under isomorphism τ, and both K and Kτ are maximal in G. Next, we consider the conditions for maximal subgroups to be isomorphic, before we study the actions of Inn(G) and Aut(G) on Ω.

For an abelian p-group, we know that G is not a complete group and Inn(G) acts on the set Ω trivially. Next, we consider the action of Aut(G) on Ω.

Research on abelian p-groups
Theorem 3.2.1

Suppose G =< g1 > × < g2 > ×...× < gm > and G is not a cyclic group with order pmn, where m ≥ 2 and n ≥ 1. Then, the maximal subgroups of G are isomorphic if and only if g1, g2,⋯ , gm have the same order pn.

Proof

When n = 1, it is obviously true. Let n > 1, because G has the form G=<g1>×<g2>××<gm>;hence,ΦG=<g1p>×<g2p>××<gmp>. G = < {g_1} > \times < {g_2} > \times \ldots \times < {g_m} > ;\;{\rm{hence,}}\;\Phi G = < g_1^p > \times < g_2^p > \times \ldots \times < g_m^p > . Let us prove adequacy. We claim that g1, g2,⋯ , gm have the same order. Otherwise, we assume that the order of g1 is ps and the order of g2 is pt, where st.

Let K1=<g1p>×<g2>×<g3>××<gm> {K_1} = < g_1^p > \times < {g_2} > \times < {g_3} > \times \ldots \times < {g_m} > . We notice that K1 is a maximal subgroup, with the type invariant (ps−1, pt, o(g3),..., o(gm)).

Let K2=<g1>×<g2p>×<g3>××<gm> {K_2} = < {g_1} > \times < g_2^p > \times < {g_3} > \times \ldots \times < {g_m} > . We find that K2 is also a maximal subgroup, with the type invariant (ps, pt−1, o(g3), ... o(gm)). Their type invariants are different, they are not isomorphic. This is a contradiction. So, the order of g1, g2,..., gm is pn.

Let us prove necessity. All the maximal subgroups of G have the next form <gip>×<a1>×<a2>××<am1> < g_i^p > \times < {a_1} > \times < {a_2} > \times \ldots \times < {a_{m - 1}} > , and gi, a1, a2,..., am−1 is a set of minimum generating system with the same order pn. Let K1 and K2 be any maximal subgroups of G. Without losing generality, let K1=<gip>×<a1>×<a2>××<am1>,K2=<gjp>×<b1>×<b2>××<bm1>. \matrix{ {{K_1} = < g_i^p > \times < {a_1} > \times < {a_2} > \times \ldots \times < {a_{m - 1}} > ,} \hfill \cr {{K_2} = < g_j^p > \times < {b_1} > \times < {b_2} > \times \ldots \times < {b_{m - 1}} > .} \hfill \cr }

We need to point out that gj, b1, b2,..., bm−1 is also a set of minimum generating system, gj, b1, b2,..., bm−1 with order pn, ij, 1 ≤ i, j ≤ m. Then, the type invariants of K1 and K2 are all (pn−1, pn,..., pn) So, K1 and K2 are isomorphic. Hence, we obtain that all maximal subgroups of G are isomorphic to each other. This completes the proof of the theorem.

Theorem 3.2.2

Assume that G is an abelian p-group and G is not cyclic. Then, P is transitive.

Proof

G has the decomposition form: G =< g1 > × < g2 > × ... × < gm >, g1, g2,..., gm with the same order pn. Assume that K1 and K2 are any maximal subgroups of G. Without losing generality, let K1=<g1p>×<g2>×<g3>××<gm> {K_1} = < g_1^p > \times < {g_2} > \times < {g_3} > \times \ldots \times < {g_m} > ; K2=<gip>×<a1>×<a2>××<am1> {K_2} = < g_i^p > \times < {a_1} > \times < {a_2} > \times \ldots \times < {a_{m - 1}} > and gi, a1, a2,..., am−1 is a set of minimum generating system with the same order pn.

Let τ : g1 → gi, g2 → a1, g3 → a2,..., gmam−1. It is easy to check that τ ∈ Aut(G) and K1τ=K2 K_1^\tau = {K_2} . Then, all maximal subgroups of G are pairwise isomorphic.

Hence, K1Aut(G)={K1τ|τAut(G)}=Ω K_1^{{\rm{Aut}}(G)} = \left\{ {K_1^\tau |\tau \in {\rm{Aut}}(G)} \right\} = \Omega , i.e. the action P of Aut(G) on Ω is transitive. This completes the proof of the theorem.

For minimal non-abelian p-groups G, Z(G) = Φ(G). For the quaternion group, Φ(G) contains element a2 with o(a2) > 1. For metacyclic groups, Φ(G) contains element ap With o(ap) > 1. For non-metacyclic groups, Φ(G) contains element c with order p. Overall, Z(G) ≠ {1}. Then, all minimal non-abelian p-groups are not complete groups. In addition, all elements in the set Ω are normal, the length of each orbit of Inn(G) is one, and the action of Inn(G) acts on the set Ω is trivial. Next, we consider the action of Aut (G) on Ω.

Research on minimal non-abelian 2-groups
Theorem 3.3.1

For quaternion group G, we have Aut(G) acting on the set Ω transitively, i.e. P is transitive.

Proof

First, Φ(G) =< a2 > and d(G) = 2. Then, G has three maximal subgroups, as follows: K1 =< a >, K2 =< b >, K3 =<ab>. Then, Ω = {K1, K2, K3}. Since these three groups are cyclic groups of order four, they are isomorphic.

Next, let us consider the orbit of G containing K1, where K1Aut(G)={K1τ|τAut(G)} K_1^{{\rm{Aut}}(G{\rm{)}}} = \left\{ {K_1^\tau |\tau \in {\rm{Aut}}(G)} \right\} . Let τ : a → b, b → ab, ab → a, a3 → a2b, a2b → a3b, a3b → a3, a2 → a2, 1 1.

We can prove τ ∈ Aut(G) easily. Then, τ2, τ3 ∈ Aut(G), τ3 is an identity transformation of G. By computation, we have K1τ=K2 K_1^\tau = {K_2} , K1τ2=K3 {\kern 1pt} K_1^{{\tau ^2}} = {K_3} , K1τ3=K1 K_1^{{\tau ^3}} = {K_1} . So, K1Aut(G)=Ω K_1^{{\rm{Aut}}(G{\rm{)}}} = \Omega . Then, P is transitive. This completes the proof of the theorem.

Theorem 3.3.2

For group M2 (n, m) in Lemma 2, if n = m ≥ 2, Aut(G) has two orbits and P is non-transitive.

Proof

(1) First, Φ(G) = < a2, b2 > and d(G) = 2. Then, G has three maximal subgroups, as follows: K1=<Φ(G),a>=<a2,b2,a>=<b2,a>;K2=<Φ(G),b>=<a2,b2,b>=<a2,b>;K3=<Φ(G),ab>=<a2,b2,ab>=<a2,ab>. \matrix{ {{K_1} = < \Phi (G),{\kern 1pt} a > = < {a^2},{\kern 1pt} {b^2},{\kern 1pt} a > = < {b^2},{\kern 1pt} a > ;} \hfill \cr {{K_2} = < \Phi (G),{\kern 1pt} b > = < {a^2},{\kern 1pt} {b^2},{\kern 1pt} b > = < {a^2},{\kern 1pt} b > ;} \hfill \cr {{K_3} = < \Phi (G),{\kern 1pt} ab > = < {a^2},{\kern 1pt} {b^2},{\kern 1pt} ab > = < {a^2},{\kern 1pt} ab > .} \hfill \cr }

Let us prove that the necessary and sufficient condition for maximal subgroups isomorphism is n = m. Since K1 and K2 are isomorphic, we claim m > 1. Conversely, if m = 1, there is b2 = 1. So, K1 =< a > and K2 =< a2, b > is non-cyclic. Then, K1 and K2 are not isomorphic. This is a contradiction.

Since K1 and K2 are isomorphic, the type invariant (2m−1, 2n) of K1 is the same as (2n−1, 2m) of K2. So, n = m. Conversely, if n = m, the type invariants of K1, K2 and K3 are all (2m−1, 2m). Then, K1, K2 and K3 are isomorphic. Then, K1, K2 and K3 are pairwise isomorphic if and only if n = m ≥ 2.

(2) Next, let us consider the orbit of Aut(G) containing K1, where K1Aut(G)={K1τ|τAut(G)}. K_1^{{\rm{Aut}}(G{\rm{)}}} = \left\{ {K_1^\tau |\tau \in {\rm{Aut}}(G)} \right\}.

According to Lemma 4, let τ : a → arbs, b → aubv, r, s, u, vN,τ ∈ Aut(G). Because an isomorphism changes a generator into a generator, aτ and bτ have the same definition relationship as a and b where rvus ≡ 1 (modulo 2). By Lemma 5, we study two cases as follows: n = m = 2 and n = m ≥ 2.

When n = m = 2, Aut(G) =< τ1, τ2, τ3, τ4 >. We obtain that K1τ1=<a3,b2>,whereτ1:aa3,bb,K1τ2=<b2,b2a>,whereτ2:ab2a,bb,K1τ3=<a,(ba)2>,whereτ3:aa,bba,K1τ4=<a,b6>,whereτ4:aa,bb3. \matrix{ {K_1^{{\tau _1}} = < {a^3},{\kern 1pt} {b^2} > ,\;{\rm{where}}\;{\tau _1}:a \to {a^3},{\kern 1pt} b \to b,} \hfill \cr {K_1^{{\tau _2}} = < {b^2},{\kern 1pt} {b^2}a > ,\;{\rm{where}}\;{\tau _2}:a \to {b^2}a,{\kern 1pt} b \to b,} \hfill \cr {K_1^{{\tau _3}} = < a,{\kern 1pt} {{(ba)}^2} > ,\;{\rm{where}}\;{\tau _3}:a \to a,{\kern 1pt} b \to ba,} \hfill \cr {K_1^{{\tau _4}} = < a,{\kern 1pt} {b^6} > ,\;{\rm{where}}\;{\tau _4}:a \to a,{\kern 1pt} b \to {b^3}.} \hfill \cr }

In fact, by computation K1τi=K1 K_1^{{\tau _i}} = {K_1} , i = 1,2,3,4. Then, K1τ=K1 K_1^\tau = {K_1} , ∀τ ∈ Aut(G). Thus, K1Aut(G)=K1 K_1^{{\rm{Aut}}(G)} = {K_1} . We obtain that K2τ1=<a6,b> K_2^{{\tau _1}} = < {a^6},{\kern 1pt} b > , K2τ2=<(b2a)2,b> {\kern 1pt} K_2^{{\tau _2}} = < ({b^2}a{)^2},{\kern 1pt} b > , K2τ3=<a2,ba> K_2^{{\tau _3}} = < {a^2},{\kern 1pt} ba > , K2τ4=<a2,b3> K_2^{{\tau _4}} = < {a^2},{\kern 1pt} {b^3} > . By computation, K2τ3=K3 K_2^{{\tau _3}} = {K_3} , K2τi=K2 {\kern 1pt} K_2^{{\tau _i}} = {K_2} , i = 1,2,4. Then, K2Aut(G)={K2,K3} K_2^{{\rm{Aut}}(G)} = \left\{ {{K_2},{K_3}} \right\} .

When n = m > 2, Aut(G) =< τ1, τ2, τ3, τ4, τ5, τ6 >. We obtain that K1τ1=<a5,b2>,whereτ1:aa5,bb,K1τ2=<a1,b2>,whereτ2:aa1,bb,K1τ3=<b2,b2a>,whereτ3:ab2a,bb,K1τ4=<a,(ba)2>,whereτ4:aa,bba,K1τ5=<a,b10>,whereτ5:aa,bb5,K1τ6=<a,b2>,whereτ6:aa,bb1 \matrix{ {K_1^{{\tau _1}} = < {a^5},{\kern 1pt} {b^2} > ,\;{\rm{where}}\;{\tau _1}:a \to {a^5},{\kern 1pt} b \to b,} \hfill \cr {K_1^{{\tau _2}} = < {a^{ - 1}},{\kern 1pt} {b^2} > ,\;{\rm{where}}\;{\tau _2}:a \to {a^{ - 1}},{\kern 1pt} b \to b,} \hfill \cr {K_1^{{\tau _3}} = < {b^2},{\kern 1pt} {b^2}a > ,\;{\rm{where}}\;{\tau _3}:a \to {b^2}a,{\kern 1pt} b \to b,} \hfill \cr {K_1^{{\tau _4}} = < a,{\kern 1pt} {{(ba)}^2} > ,\;{\rm{where}}\;{\tau _4}:a \to a,{\kern 1pt} b \to ba,} \hfill \cr {K_1^{{\tau _5}} = < a,{\kern 1pt} {b^{10}} > ,\;{\rm{where}}\;{\tau _5}:a \to a,{\kern 1pt} b \to {b^5},} \hfill \cr {K_1^{{\tau _6}} = < a,{\kern 1pt} {b^{ - 2}} > ,\;{\rm{where}}\;{\tau _6}:a \to a,{\kern 1pt} b \to {b^{ - 1}}} \hfill \cr }

In fact, by computation, K1τi=K1 K_1^{{\tau _i}} = {K_1} , i = 1,2,3,4,5,6. Then, K1τ=K1 K_1^\tau = {K_1} , ∀τ ∈ Aut(G). Thus, K1Aut(G)=K1 K_1^{{\rm{Aut}}(G)} = {K_1} .

We obtain that K2τ1=<a10,b> K_2^{{\tau _1}} = < {a^{10}},{\kern 1pt} b > , K2τ2=<a2,b> {\kern 1pt} K_2^{{\tau _2}} = < {a^{ - 2}},{\kern 1pt} b > , K2τ3=<(b2a)2,b> {\kern 1pt} K_2^{{\tau _3}} = < ({b^2}a{)^2},{\kern 1pt} b > , K2τ4=<a2,ba> {\kern 1pt} K_2^{{\tau _4}} = < {a^2},{\kern 1pt} ba > , K2τ5=<a2,b5> {\kern 1pt} K_2^{{\tau _5}} = < {a^2},{\kern 1pt} {b^5} > , K2τ6=<a2,b1> K_2^{{\tau _6}} = < {a^2},{\kern 1pt} {b^{ - 1}} > . By computation, K2τi=K2 K_2^{{\tau _i}} = {K_2} , i = 1,2,3,5,6. Since ab = a1+2n−1, i.e. b−1ab = aa2n−1, then, ab = baa2n−1 and ba = aba−2n−1. Hence, by computation, K2τ4=<a2,ba>=<a2,aba2n1>=<a2,ab>=K3 K_2^{{\tau _4}}{\kern 1pt} = {\kern 1pt} < {a^2},{\kern 1pt} ba > {\kern 1pt} = {\kern 1pt} < {a^2},{\kern 1pt} ab{a^{ - {2^{n - 1}}}} > {\kern 1pt} = {\kern 1pt} < {a^2},{\kern 1pt} ab > {\kern 1pt} = {\kern 1pt} {K_3} .

Then, K2Aut(G)={K2,K3} K_2^{{\rm{Aut}}(G)} = \{ {K_2},{\kern 1pt} {K_3}\} .

Therefore, the length of the orbit containing K1 is one and the length of the orbit containing K2 is two. Thus, Aut(G) does not act on the set Ω transitively. This completes the proof of the theorem.

Theorem 3.3.3

For group M2(n, m, 1), when n = m ≥ 2, P is transitive.

Proof

First, Φ(G) =< a2, b2, c > and d(G) = 2. Then, G has three subgroups that are maximal, as follows: K1=<Φ(G),a>=<a,b2,c>,K2=<Φ(G),b>=<b,a2,c>andK3=<Φ(G),ab>=<a2,b2,c,ab>=<ab,b2,c>. \matrix{ {{K_1} = < \Phi (G),{\kern 1pt} a > = < a,{\kern 1pt} {b^2},{\kern 1pt} c > ,{\kern 1pt} {K_2}{\kern 1pt} = {\kern 1pt} < \Phi (G),{\kern 1pt} b > = < b,{\kern 1pt} {a^2},{\kern 1pt} c > {\rm{and}}} \hfill \cr {{K_3} = < \Phi (G),{\kern 1pt} ab > = < {a^2},{\kern 1pt} {b^2},{\kern 1pt} c,{\kern 1pt} ab > = < ab,{\kern 1pt} {b^2},{\kern 1pt} c > .} \hfill \cr }

Then, Ω = {K1, K2, K3}.

Let us prove that the necessary and sufficient condition for maximal subgroups isomorphism is n = m. Note that the abelian two-group K1 has type invariants (2n, 2m−1, 2) and the abelian two-group K2 has type invariants (2m, 2n−1, 2). Since K1 and K2 are isomorphic, we have n = m. Conversely, if n = m, the type invariants of K1, K2 and K3 are the same. So, they are isomorphic.

Thus, K1, K2 and K3 are pairwise isomorphic if and only if n = m ≥ 2.

Next, let us consider the orbit of Aut(G) containing K1, where K1Aut(G)={K1σ|σAut(G)} K_1^{{\rm{Aut}}(G)} = \left\{ {K_1^\sigma {\kern 1pt} |{\kern 1pt} \sigma \in {\rm{Aut}}(G)} \right\}

Letting σ1 be an identical transformation, K1σ1=K1 K_1^{{\sigma _1}} = {K_1} . Letting σ2 : a → ab, b → b, by an easy computation, σ2 ∈ Aut(G) and K1σ2=K3 K_1^{{\sigma _2}} = {K_3} . Letting σ3 : a → b, b → a, we have σ3 ∈ Aut(G) and K1σ3=K2 K_1^{{\sigma _3}} = {K_2} . Then, K1Aut(G)=Ω K_1^{{\rm{Aut}}(G)} = \Omega . Therefore, P is transitive. This completes the proof of the theorem.

Research on group Mp (n, m) with p>2
Theorem 3.4.1

For group Mp (n, m) with n = m ≥ 2, P is non-transitive.

Proof

First, d(G) = 2 and Φ(G) =< ap, bp >. Then, G has p + 1 subgroups that are maximal, as follows: K1=<Φ(G),a>=<bp,a>,K2=<Φ(G),b>=<ap,b>,Ms=<Φ(G),abs>s=1,2,,p1. \matrix{ {{K_1} = < \Phi (G),{\kern 1pt} a > = < {b^p},{\kern 1pt} a > ,} \hfill \cr {{K_2} = < \Phi (G),{\kern 1pt} b > = < {a^p},{\kern 1pt} b > ,} \hfill \cr {{M_s} = < \Phi (G),{\kern 1pt} a{b^s} > s = 1,2, \ldots ,p - 1.} \hfill \cr } Then, Ω = {K1, K2, M1,..., Mp−1}.

We can prove that K1, K2, M1,..., Mp−1 are pairwise isomorphic if and only if n = m ≥ 2.

Let τ : a → arbs, b → aubv, r, s, u, vN, τ ∈ Aut(G). Because an isomorphism changes a generator into a generator, aτ and bτ have the same definition relationship as a and b, where prvus. By Lemma 5, we only consider that τ1 : a → ap+1, b → b, τ2 : a → bpa, b → b, τ3 : a → a, b → ba, τ4 : a → a, b → b1+p, τ5 : a → as0, b → b, s0 = tpn−1, t is the primitive root of modulo pn.

Then, K1τ1=<a1+p,b> K_1^{{\tau _1}} = < {a^{1 + p}},{\kern 1pt} b > , K1τ2=<bpa,bp> {\kern 1pt} K_1^{{\tau _2}} = < {b^p}a,{\kern 1pt} {b^p} > , K1τ3=<a,(ba)p> K_1^{{\tau _3}} = < a,{\kern 1pt} {(ba)^p} > , K1τ4=<a,bp(1+p)> K_1^{{\tau _4}} = < a,{\kern 1pt} {b^{p(1 + p)}} > , K1τ5=<as0,bp> K_1^{{\tau _5}} = < {a^{{s_0}}},{\kern 1pt} {b^p} > . In fact, by computation, K1τi=K1 K_1^{{\tau _i}} = {K_1} , i = 1,⋯ , 5. So, K1τ=K1 K_1^\tau = {K_1} , ∀τ ∈ Aut(G). Thus, K1Aut(G)=K1 K_1^{{\rm{Aut}}(G)} = {K_1} . So, Aut(G) does not act on the set Ω transitively. This completes the proof of the theorem.

Theorem 3.4.2

For group Mp (n, m, 1), when m = n ≥ 2, P is transitive.

Proof

First, Φ(G) =< c, ap, bp > and d(G) = 2. So, G has p + 1 maximal subgroups, as follows:

K =< Φ(G), a >=< a, bp, c >, Mi =< Φ(G), asb >=< ap, asb, bp, c >, s = 0,1,⋯ , p − 1. Then, we get Ω = {K, M0,⋯ , Mp−1}.

We can prove that K, M0,⋯ , Mp−1 are pairwise isomorphic if and only if n = m ≥ 2. Let σ ∈ Aut(G), σ : a → asb, b → aubv, u = sv + kp, u, v, kN, s = 1,⋯ , p − 1. By computation, Kσ = Ms, s = 1,⋯ , p − 1.

We can also let τ ∈ Aut(G), τ : a → b, b → a. By computation, Kσ = Ms, s = 0. Letting τ be an identical transformation, Kτ = K. Then, K1Aut(G)=Ω K_1^{{\rm{Aut}}(G)} = \Omega . Therefore, P is transitive. This completes the proof of the theorem.

Conclusion

In this paper, for Aut(G) of finite abelian p-groups and minimal non-abelian p-groups, we study their actions on Ω. We get some results about them. When G is a non-cyclic abelian p-group, we obtain that the action P of Aut(G) on Ω is transitive. For a quaternion group, Aut(G) has only one orbit and P is transitive; when G is a non-metacyclic group, which is minimal non-abelian, the result is the same as for the quaternion group. When G is a metacyclic group, which is minimal non-abelian, P is non-transitive.

Redei L. Das schiefe product in der gruppentheorie. Comment Math Helvet, 1947, 20, pp. 25–267. RedeiL Das schiefe product in der gruppentheorie Comment Math Helvet 1947 20 25 267 10.1007/BF02568131 Search in Google Scholar

Hermann P Z. On finite p-groups with isomorphic maximal subgroups. Austral Math Soc, 1990, 48(2), pp. 199–213. HermannP Z On finite p-groups with isomorphic maximal subgroups Austral Math Soc 1990 48 2 199 213 10.1017/S1446788700035631 Search in Google Scholar

Mann A. On p-groups whose isomorphic maximal subgroup are isomorphic. Austral Math Soc, 1995, 59(2), pp. 143–147. MannA On p-groups whose isomorphic maximal subgroup are isomorphic Austral Math Soc 1995 59 2 143 147 10.1017/S1446788700038556 Search in Google Scholar

Wangne Nedelar, Huk. Regular dessins uniquely detemined by a nilpotent automorphism group. Group Theory, 2018, 21(3), pp. 397–415. Wangne Nedelar Huk Regular dessins uniquely detemined by a nilpotent automorphism group Group Theory 2018 21 3 397 415 10.1515/jgth-2017-0044 Search in Google Scholar

Guining. Ban, Jianping. Wu, Yu. Zhang, Zhongjian. Zhang. The order of automorphism group of a special finite p group. Journal of Yunnan University, 2008, 30, pp. 215–219. BanGuining. WuJianping. ZhangYu. ZhangZhongjian. The order of automorphism group of a special finite p group Journal of Yunnan University 2008 30 215 219 Search in Google Scholar

Guining. Ban, Xinzheng. Zhang, Yong. Wang. The order of automorphism group of p group, Journal of Southwest Normal University. 2005, 4(30), pp. 600–603. BanGuining. ZhangXinzheng. WangYong. The order of automorphism group of p group Journal of Southwest Normal University 2005 4 30 600 603 Search in Google Scholar

Changcheng. Xiao. On the order of automorphism group of p group, Journal of Huaqiao University. 1986, 1(17), pp. 13–20. XiaoChangcheng. On the order of automorphism group of p group Journal of Huaqiao University 1986 1 17 13 20 Search in Google Scholar

G.T. Helleloid. A survey on automorphism groups of finite p-groups, arXiv:math/0610294v2 [math. GR] 25 Oct 2006. HelleloidG.T. A survey on automorphism groups of finite p-groups arXiv:math/0610294v2 [math. GR] 25 Oct 2006 Search in Google Scholar

Mingyao. Xu. Finite group guidance (Volume I). Beijing: Kexue Press, 2001. XuMingyao. Finite group guidance (Volume I) Beijing Kexue Press 2001 Search in Google Scholar

Jin Ho Kwak, Mingyao Xu. Finite group theory for combinatorics (Volume one). Combinatorial and Computational Mathimatics Center Pohang University of Science and Technology, 2005. KwakJin Ho XuMingyao Finite group theory for combinatorics (Volume one) Combinatorial and Computational Mathimatics Center Pohang University of Science and Technology 2005 Search in Google Scholar

Jing. Xu, Mingyao. Xu. A preliminary study of modern algebra. Beijing: Beijing University Press, 2020. XuJing. XuMingyao. A preliminary study of modern algebra Beijing Beijing University Press 2020 Search in Google Scholar

Qinhai. Zhang, Lijian. An. Construction of finite p groups (Volume I). Beijing: Kexue Press, 2017, pp. 16–17. ZhangQinhai. AnLijian. Construction of finite p groups (Volume I) Beijing Kexue Press 2017 16 17 Search in Google Scholar

Rong. Chen. The structure of the automorphism group of the subcyclic inner commutative group. ShanXi: Shanxi Normal University, 2007. ChenRong. The structure of the automorphism group of the subcyclic inner commutative group ShanXi Shanxi Normal University 2007 Search in Google Scholar

Yuanda. Zhang. Construction of finite groups (Volume I). Beijing: Kexue Press, 2019, pp. 142–143. ZhangYuanda. Construction of finite groups (Volume I) Beijing Kexue Press 2019 142 143 Search in Google Scholar

Jingen. Yang. Lecture notes on modern algebra. Beijing: Kexue Press, 2021, pp. 75–76. YangJingen. Lecture notes on modern algebra Beijing Kexue Press 2021 75 76 Search in Google Scholar

Herui. Zhang. Fundamentals of modern algebra. Beijing: Higher Education Press, 2012, pp. 56–60. ZhangHerui. Fundamentals of modern algebra Beijing Higher Education Press 2012 56 60 Search in Google Scholar

B. Huppert, Translated by Jiang Hao, Yu Shuxia. Finite group theory. FuZhou: Fujian People's Press, 1992. HuppertB. Translated by JiangHao YuShuxia Finite group theory FuZhou Fujian People's Press 1992 Search in Google Scholar

Efang. Wang. Fundamentals of finite group theory. Beijing: QingHua University Press, 2012. WangEfang. Fundamentals of finite group theory Beijing QingHua University Press 2012 Search in Google Scholar

Mingyao. Xu, Haipeng. Qu. Finite p Group. Beijing: Beijing University Press, 2010, pp. 10–12. XuMingyao. QuHaipeng. Finite p Group Beijing Beijing University Press 2010 10 12 Search in Google Scholar

Mingyao. Xu. Finite group preliminary. Beijing: Kexue Press, 2014, pp. 26–29. XuMingyao. Finite group preliminary Beijing Kexue Press 2014 26 29 Search in Google Scholar

Zhongmu. Chen. Inner outer-group and minimal non-group. Chongqing: Southwest Normal University Press, 1988. ChenZhongmu. Inner outer-group and minimal non-group Chongqing Southwest Normal University Press 1988 Search in Google Scholar

Wujie. Shi, Shiheng. Li. Guidance of finite group theory. Beijing: Kexue Press, 2019, pp. 44–45. ShiWujie. LiShiheng. Guidance of finite group theory Beijing Kexue Press 2019 44 45 Search in Google Scholar

Xi. Guo, Guikang. Wu. Modern algebra. Shanghai: East China Normal University Press, 2019, pp. 15–16. GuoXi. WuGuikang. Modern algebra Shanghai East China Normal University Press 2019 15 16 Search in Google Scholar

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