Groups of Prime Power Order. Volume 6

Yakov G. Berkovich, Zvonimir Janko

• Publisher: De Gruyter
• Year: 2018
• Print ISBN:
  9783110530971
• eBook ISBN:
  9783110531008
• Topic: Mathematics
• Subtopic: Group Theory
• Index: Mathematics

This is the sixth volume of a comprehensive and elementary treatment of finite group theory. This volume contains many hundreds of original exercises (including solutions for the more difficult ones) and an extended list of about 1000 open problems. The current book is based on Volumes 1–5 and it is suitable for researchers and graduate students working in group theory.

§ 263Nonabelian 2-groups G with CG(x) ≤ H for all H ∈ Γ1 and x ∈ H − Z(G)

In this section we solve Problem 3702 for p = 2 and prove the following result.

Theorem 263.1. Let G be a nonabelian 2-group with CG(x) ≤ H for all H ∈ Γ1 and x ∈ H − Z(G). Then we have Φ(G) = Z(G) and each maximal abelian subgroup of G has order 2|Z(G)|.

Conversely, all such nonabelian 2-groups satisfy the assumptions of our theorem.

Proof. Let G be a nonabelian 2-group with CG(x) ≤ H for all H ∈ Γ1 and x ∈ H − Z(G).

Let H be a fixed maximal subgroup in G. Then Z(G) < H so that Z(G) ≤ Φ(G). For each g ∈ G − H, g2 ∈ Z(G) (indeed, CG(g2) ≰ H) and so by a result of Burnside applied to G/Z(G), we have that H/Z(G) is abelian and g inverts each element of H/Z(G) (indeed, all elements in the set G/Z(G) − H/Z(G) have order 2). Hence each maximal subgroup of G/Z(G) is abelian and so we have either G' ≤ Z(G) (and then cl(G) = 2) or G/Z(G) is minimal nonabelian. Suppose that we have the second case. Since minimal nonabelian (two generator) group G/Z(G) is generated with involutions, we get G/Z(G) ≅ D8 and since g inverts each element of H/Z(G), it follows that H/Z(G) ≅ C4 so that H is abelian and G is not of class 2. But H was an arbitrary maximal subgroup in G so that each maximal subgroup of G is abelian. This implies that |G'| = 2 and so G' ≤ Z(G) and G is of class 2, a contradiction.

We have proved that G is of class 2. Since G/Z(G) is abelian and is generated by involutions, G/Z(G) is elementary abelian and so Φ(G) = ℧1(G) ≤ Z(G). But we have also Z(G) ≤ Φ(G) and so Φ(G) = Z(G).

Suppose that G possesses an abelian subgroup A > Z(G) such that A/Z(G) ≅ E4. Then A < G. Let K be a maximal subgroup of G which does not contain A. Then (A ∩ K) > Z(G) and if x ∈ (A ∩ K) − Z(G), then CG(x) ≰ K since A ≤ CG(a), a contradiction. We have proved that each maximal abelian subgroup in G is of order 2|Z(G)|. The theorem is proved since the converse is clear. $\square$

§ 264Nonabelian 2-groups of exponent ≥ 16 all of whose minimal nonabelian subgroups, except one, have order 8

In § 90 we have determined the nonabelian 2-groups all of whose minimal nonabelian subgroups are of order 8, i.e., they are isomorphic to D8 or Q8. According to an idea of the first author, we study here such 2-groups G...

Table of contents

• Cover
• Title Page
• Copyright
• Contents
• List of definitions and notations
• Preface
• § 257 Nonabelian p-groups with exactly one minimal nonabelian subgroup of exponent > p
• § 258 2-groups with some prescribed minimal nonabelian subgroups
• § 259 Nonabelian p-groups, p > 2, all of whose minimal nonabelian subgroups are isomorphic to Mp3
• § 260 p-groups with many modular subgroups Mpn
• § 261 Nonabelian p-groups of exponent > p with a small number of maximal abelian subgroups of exponent > p
• § 262 Nonabelian p-groups all of whose subgroups are powerful
• § 263 Nonabelian 2-groups G with CG(x) ≤ H for all H ∈ Γ1 and x ∈ H − Z(G)
• § 264 Nonabelian 2-groups of exponent ≥ 16 all of whose minimal nonabelian subgroups, except one, have order 8
• § 265 p-groups all of whose regular subgroups are either absolutely regular or of exponent p
• § 266 Nonabelian p-groups in which any two distinct minimal nonabelian subgroups with a nontrivial intersection are non-isomorphic
• § 267 Thompson’s A × B lemma
• § 268 On automorphisms of some p-groups
• § 269 On critical subgroups of p-groups
• § 270 p-groups all of whose Ak-subgroups for a fixed k > 1 are metacyclic
• § 271 Two theorems of Blackburn
• § 272 Nonabelian p-groups all of whose maximal abelian subgroups, except one, are either cyclic or elementary abelian
• § 273 Nonabelian p-groups all of whose noncyclic maximal abelian subgroups are elementary abelian
• § 274 Non-Dedekindian p-groups in which any two nonnormal subgroups normalize each other
• § 275 Nonabelian p-groups with exactly p normal closures of minimal nonabelian subgroups
• § 276 2-groups all of whose maximal subgroups, except one, are Dedekindian
• § 277 p-groups with exactly two conjugate classes of nonnormal maximal cyclic subgroups
• § 278 Nonmetacyclic p-groups all of whose maximal metacyclic subgroups have index p
• § 279 Subgroup characterization of some p-groups of maximal class and close to them
• § 280 Nonabelian p-groups all of whose maximal subgroups, except one, are minimal nonmetacyclic
• § 281 Nonabelian p-groups in which any two distinct minimal nonabelian subgroups have a cyclic intersection
• § 282 p-groups with large normal closures of nonnormal subgroups
• § 283 Nonabelian p-groups with many cyclic centralizers
• § 284 Nonabelian p-groups, p > 2, of exponent > p2 all of whose minimal nonabelian subgroups are of order p3
• § 285 A generalization of Lemma 57.1
• § 286 Groups ofexponent p with many normal subgroups
• § 287 p-groups in which the intersection of any two nonincident subgroups is normal
• § 288 Nonabelian p-groups in which for every minimal nonabelian M < G and x ∈ G − M, we have CM(x) = Z(M)
• § 289 Non-Dedekindian p-groups all of whose maximal nonnormal subgroups are conjugate
• § 290 Non-Dedekindian p-groups G with a noncyclic proper subgroup H such that each subgroup which is nonincident with H is normal in G
• § 291 Nonabelian p-groups which are generated by a fixed maximal cyclic subgroup and any minimal nonabelian subgroup
• § 292 Nonabelian p-groups generated by any two non-conjugate minimal nonabelian subgroups
• § 293 Exercises
• § 294 p-groups, p > 2, whose Frattini subgroup is nonabelian metacyclic
• § 295 Any irregular p-group contains a non-isolated maximal regular subgroup
• § 296 Non-Dedekindian p-groups all of whose nonnormal maximal cyclic subgroups are C-equivalent
• § 297 On 2-groups without elementary abelian subgroup of order 8
• § 298 Non-Dedekindian p-groups all of whose subgroups of order ≤ ps (s ≥ 1 fixed) are normal
• § 299 On p'-automorphisms of p-groups
• § 300 On p-groups all of whose maximal subgroups of exponent p are normal and have order pp
• § 301 p-groups of exponent > p containing < p maximal abelian subgroups of exponent > p
• § 302 Alternate proof of Theorem 109.1
• § 303 Nonabelian p-groups of order > p4 all of whose subgroups of order p4 are isomorphic
• § 304 Non-Dedekindian p-groups in which each nonnormal subgroup has a cyclic complement in its normalizer
• § 305 Nonabelian p-groups G all of whose minimal nonabelian subgroups M satisfy Z(M) ≤ Z(G)
• § 306 Nonabelian 2-groups all of whose maximal subgroups, except one, are quasi-Hamiltonian or abelian
• § 307 Nonabelian p-groups, p > 2, all of whose maximal subgroups, except one, are quasi-Hamiltonian or abelian
• § 308 Nonabelian p-groups with an elementary abelian intersection of any two distinct maximal abelian subgroups
• § 309 Minimal non-p-central p-groups
• § 310 Nonabelian p-groups in which each element in any minimal nonabelian subgroup is half-central
• § 311 Nonabelian p-groups G of exponent p in which CG(x) = ⟨x⟩G for all noncentral x ∈ G
• § 312 Nonabelian 2-groups all of whose minimal nonabelian subgroups, except one, are isomorphic to M2(2, 2) = ⟨a, b | a4 = b4 = 1, ab = a−1⟩
• § 313 Non-Dedekindian 2-groups all of whose maximal Dedekindian subgroups have index 2
• § 314 Theorem of Glauberman–Mazza on p-groups with a nonnormal maximal elementary abelian subgroup of order p2
• § 315 p-groups with some non-p-central maximal subgroups
• § 316 Nonabelian p-groups, p > 2, of exponent > p3 all of whose minimal nonabelian subgroups, except one, have order p3
• § 317 Nonabelian p-groups, p > 2, all of whose minimal nonabelian subgroups are isomorphic to Mp(2, 2)
• § 318 Nonabelian p-groups, p > 2, of exponent > p2 all of whose minimal nonabelian subgroups, except one, are isomorphic to Mp(2, 2)
• § 319 A new characterization of p-central p-groups
• § 320 Nonabelian p-groups with exactly one non-p-central minimal nonabelian subgroup
• § 321 Nonabelian p-groups G in which each element in G − Φ(G) is half-central
• § 322 Nonabelian p-groups G such that CG(H) = Z(G) for any nonabelian H ≤ G
• § 323 Nonabelian p-groups that are not generated by its noncyclic abelian subgroups
• § 324 A separation of metacyclic and nonmetacyclic minimal nonabelian subgroups in nonabelian p-groups
• § 325 p-groups which are not generated by their nonnormal subgroups, 2
• § 326 Nonabelian p-groups all of whose maximal abelian subgroups are normal
• A.110 Non-absolutely regular p-groups all of whose maximal absolutely regular subgroups have index p
• A.111 Nonabelian p-groups of exponent > p all of whose maximal abelian subgroups of exponent > p are isolated
• A.112 Metacyclic p-groups with an abelian maximal subgroup
• A.113 Nonabelian p-groups with a cyclic intersection of any two distinct maximal abelian subgroups
• A.114 An analog of Thompson’s dihedral lemma
• A.115 Some results from Thompson’ papers and the Odd Order paper
• A.116 On normal subgroups of a p-group
• A.117 Theorem of Mann
• A.118 On p-groups with given isolated subgroups
• A.119 Two-generator normal subgroups of a p-group G that contained in Φ(G) are metacyclic
• A.120 Alternate proofs of some counting theorems
• A.121 On p-groups of maximal class
• A.122 Criteria of regularity
• A.123 Nonabelian p-groups in which any two nonincident subgroups have an abelian intersection
• A.124 Characterizations of the p-groups of maximal class and the primary ECF-groups
• A.125 Nonabelian p-groups all of whose proper nonabelian subgroups have exponent p
• A.126 On p-groups with abelian automorphism groups
• A.127 Alternate proof of Proposition 1.23
• A.128 Alternate proof of the theorem of Passman on p-groups all of whose subgroups of order ≤ ps (s ≥ 1 is fixed) are normal
• A.129 Alternate proofs of Theorems 309.1 and 309.2 on minimal non-p-central p-groups
• A.130 Non-Dedekindian p-groups all of whose nonnormal maximal cyclic subgroups are conjugate
• A.131 A characterization of some 3-groups of maximal class
• A.132 Alternate approach to classification of minimal non-p-central p-groups
• A.133 Nonabelian p-groups all of whose minimal nonabelian subgroups are isomorphic to Mp(n, n) or Mp(n, n, 1) for a fixed natural n > 1
• A.134 On irregular p-groups G = Ω1(G) without subgroup of order pp+1 and exponent p
• A.135 Nonabelian 2-groups of given order with maximal possible number of involutions
• A.136 On metacyclic p-groups
• A.137 Alternate proof of Lemma 207.1
• A.138 Subgroup characterization of a p-group of maximal class with an abelian subgroup of index p
• Research problems and themes VI
• Bibliography
• Author index
• Subject index