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Unformatted text preview: GLASNIK MATEMATI ˇ CKI Vol. 41(61)(2006), 239 – 258 SHORT PROOFS OF SOME BASIC CHARACTERIZATION THEOREMS OF FINITE p-GROUP THEORY Yakov Berkovich University of Haifa, Israel Abstract. We offer short proofs of such basic results of finite p-group theory as theorems of Blackburn, Huppert, Ito-Ohara, Janko, Taussky. All proofs of those theorems are based on the following result: If G is a nonabelian metacyclic p-group and R is a proper G-invariant subgroup of G , then G/R is not metacyclic. In the second part we use Blackburn’s theory of p-groups of maximal class. Here we prove that a p-group G is of maximal class if and only if Ω * 2 ( G ) = h x ∈ G | o ( x ) = p 2 i is of maximal class. We also show that a noncyclic p-group G of exponent > p contains two distinct maximal cyclic subgroups A and B of orders > p such that | A ∩ B | = p , unless p = 2 and G is dihedral. 1 ◦ . This note is a continuation of the author’s previous papers [Ber1, Ber2, Ber4]. Only finite p-groups, where p is a prime, are considered. The same notation as in [Ber1] is used. The n th member of the lower central se- ries of G is denoted by K n ( G ). Given a p-group G and a natural num- ber n , set f n ( G ) = h x p n | x ∈ G i , Ω n ( G ) = h x ∈ G | o ( x ) ≤ p n i , Ω * n ( G ) = h x ∈ G | o ( x ) = p n i , f 2 ( G ) = f 1 ( f 1 ( G )), p d( G ) = | G : Φ( G ) | , where Φ( G ) is the Frattini subgroup of G . Next, G is the derived subgroup and Z( G ) is the center of G . A group G of order p m is of maximal class if m > 2 and cl( G ) = m- 1. A group G is metacyclic if it contains a normal cyclic subgroup C such that G/C is cyclic. A group G is said to be minimal nonabelian if it is nonabelian but all its proper subgroups are abelian. A p- group G is regular if, for x, y ∈ G , there is z ∈ h x, y i such that ( xy ) p = x p y p z p . A p-group G is absolutely regular if | G/ f 1 ( G ) | < p p . A p-group G is powerful 2000 Mathematics Subject Classification. 20D15. Key words and phrases. Finite p-groups, metacyclic p-groups, minimal nonabelian p-groups, p-groups of maximal class, regular and absolutely regular p-groups, powerful p- groups. 239 240 Y. BERKOVICH [LM] provided G ≤ f p ( G ), where 2 = 2 and p = 1 for p > 2. By c n ( G ) we denote the number of cyclic subgroups of order p n in G . In Section 2 ◦ we show that some basic results of p-group theory are easy consequences of Theorem 2. In Section 3 ◦ we use [Ber1, Theorem 5.1] (= Lemma 1(d)), a variant of Blackburn’s result [Ber3, Theorem 9.7], charac- terizing p-groups of maximal class. The following results of this note are new: Supplement 2 to Corollary 11, Corollary 14, theorems 22, 24, 25, 27, 30, Supplements 1 and 2 to Theorem 22 and Remark 18....
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