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• W1598346960 abstract "We study the probability of a given element, in the commutator subgroup of a group, to be equal to a commutator of two randomly chosen group elements, and compute explicit formulas for calculating this probability for some interesting classes of groups having only two different conjugacy class sizes. We re-prove the fact that if G is a finite group such that the set of its conjugacy class sizes is {1,p}, where p is a prime integer, then G is isoclinic (in the sense of P. Hall) to an extraspecial p-group. Notice that for a given element g ∈ G, Prg(G) measures the probability that the commutator of two randomly chosen group elements is equal to g. Obviously, when g = 1, Prg(G) = Pr(G). Pournaki and Sobhani (12) mainly studied Prg(G) for finite groups G which have only two different irreducible complex character degrees and obtained an impressive formula for Prg(G) for such groups G. In particular, using character theoretic techniques, they obtained explicit formulas for Prg(G), when G is a finite group with |2(G)| = p, where p is a prime integer and 2(G) denotes the commutator subgroup of G. In this situation, there can only be two cases, namely, (i) 2(G) ≤ Z(G) or (ii) 2(G) ∩ Z(G) = 1, where Z(G) denotes the center of G. These are the cases which were studied by Rusin (13) in 1979 and explicit formulas were ontained for Pr(G). For finite groups G which have only two different irreducible complex character degrees 1 and m (say), they proved (12, Theorem 2.2) that for each 1 6 g ∈ K(G), (1.1) Prg(G) = (1/|2(G)|)(1 − 1/m 2 ), where K(G) denotes the set of all commutators in G. We assume that 1 ∈ K(G). Motivated by the results of Rusin (13), and Pournaki and Sobhani (12), we investigate Prg(G) for some classes of finite groups G which have only two different conjugacy class sizes. Explicit formulas are obtained for some interesting classes of finite groups using purely group theoretic techniques. A finite group G is said to be a Camina group if x G = x2(G) for all" @default.
• W1598346960 created "2016-06-24" @default.
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• W1598346960 date "2012-12-18" @default.
• W1598346960 modified "2023-09-27" @default.
• W1598346960 title "PROBABILITY THAT A GIVEN ELEMENT OF A GROUP IS A COMMUTATOR OF ANY TWO RANDOMLY CHOSEN GROUP ELEMENTS" @default.
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