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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 pGROUP THEORY Yakov Berkovich University of Haifa, Israel Abstract. We offer short proofs of such basic results of finite pgroup theory as theorems of Blackburn, Huppert, ItoOhara, Janko, Taussky. All proofs of those theorems are based on the following result: If G is a nonabelian metacyclic pgroup and R is a proper Ginvariant subgroup of G , then G/R is not metacyclic. In the second part we use Blackburn’s theory of pgroups of maximal class. Here we prove that a pgroup 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 pgroup 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 pgroups, 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 pgroup 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 pgroup G is absolutely regular if  G/ f 1 ( G )  < p p . A pgroup G is powerful 2000 Mathematics Subject Classification. 20D15. Key words and phrases. Finite pgroups, metacyclic pgroups, minimal nonabelian pgroups, pgroups of maximal class, regular and absolutely regular pgroups, 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 pgroup 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 pgroups 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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 Winter '11
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