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Conjugacy structure type and degree structure type in finite p-groups

Let G be a finite p−group, and denote by k(G) number of conjugacy classes in G. The aim of this paper is to introduce the conjugacy structure type and degree structure type for p−groups, and determine these parameters for p−groups of order p5 and calculate k(G) for them. | Turk J Math 24 (2000) , 321 – 326. ¨ ITAK ˙ c TUB Conjugacy Structure Type and Degree Structure Type in Finite p−groups Yadolah Marefat Abstract Let G be a finite p−group, and denote by k(G) number of conjugacy classes in G. The aim of this paper is to introduce the conjugacy structure type and degree structure type for p−groups, and determine these parameters for p−groups of order p5 , and calculate k(G) for them. Key Words: breadth, conjugacy structure type, degree structure type. 1. Introduction Let G be a finite p−group, and denote by k(G) number of conjugacy classes of G. We remind the reader that an element g of p−group G is said to have breadth bG(g)(b(g) if no ambiguity is possible) if pbG(g) is the size of conjugacy class of g in G. The breadth b(G) of G will be maximum of breadths of its elements. We have, b(G) = 1 if and only if |G0 | = p (see [4]), b(G) = 2 if and only if |G0 | = p2 or |G : Z(G)| = p3 and |G0 | = p3 (see [7]). Let si be the number of conjugacy classes of size pi in G. Let m be a Pm non-negative integer such that sm 6= 0, and si = 0 for i > m. Then |G| = i=0 si pi , and P k(G) = m i=0 si . We define the tuple (s0 , s1 , . . . , sm ), Conjugacy Structure Type of G, Definition 1. and denote by Tc (G). It is clear that G is abelian if and only if m = 0. 321 MAREFAT Let αi be the number of irreducible characters of G of order pi . Let h be Ph a non-negative integer such that αh 6= 0, and αi = 0 for i > h. Then |G| = i=0 αi p2i , P and k(G) = hi=0 αi . We define the tuple (α0 , α1, . . . , αh ), Degree Structure Type of G, Defintion 2. and denote by Td (G). We know that b(G) is the maximum index of i such that si is nonzero, that means b(G) = m. We denote by β(G) the maximum index of i such that αi is nonzero that is β(G) = h. Burnside’s Formula. Let G be a finite p−group and M be a maximal subgroup in G. If s and t are the number respectively of invariant and fused conjugacy classes of M then k(G) = ps + t p = s(p − p1 ) + k(M .