In this article, we determine all groups with exactly ten element centralizers. Also we obtain the maximum size of the pairwise non-commuting elements of such groups.
In this article, we determine the structure of all nonabelian groups G such that G has the minimum number of the element centralizers among nonabelian groups of the same order. As an application of this result, we obtain the sharp lower bound for ω(G) in terms of the order of G where ω(G) is the maximum size of a set of the pairwise noncommuting elements of G.
For a group G, let cent(G) denote the set of centralizers of single elements of G and nacent(G) denote the set of all nonabelian centralizers belonging to cent(G). We first characterize all finite groups G with |nacent(G)| = 2. We denote by ω(G), the maximum possible size of a subset of pairwise noncommuting elements of a finite group G. In this article we find a necessary and sufficient condition for some finite groups G satisfying |cent(G)| = |nacent (G)| + ω(G). In particular we show that this equality is valid for some simple groups.
In this paper, we find the number of the element centralizers of a finite group [Formula: see text] such that the central factor of [Formula: see text] is the projective special linear group of degree 2 or the Suzuki group. Our results generalize some main results of [Ashrafi and Taeri, On finite groups with a certain number of centralizers, J. Appl. Math. Comput. 17 (2005) 217–227; Schmidt, Zentralisatorverbände endlicher Gruppen, Rend. Sem. Mat. Univ. Padova 44 (1970) 97–131; Zarrin, On element centralizers in finite groups, Arch. Math. 93 (2009) 497–503]. Also, we give an application of these results.
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