On 10-Centralizer Groups of Odd Order

Let be a group, and let denote the number of distinct centralizers of its elements. A group is called -centralizer if . In this paper, we investigate the structure of finite groups of odd order with and prove that there is no finite nonabelian group of odd order with . 1. Introduction Throughout this paper all groups mentioned are assumed to be finite, and we will use usual notation; for example, denotes the cyclic group of order , and denotes the semidirect product of and with normal subgroup , where is a positive integer and is a prime. For a group , denotes the center of , and , where is the centralizer of the element in ; that is, . A group is a -group if is abelian for every . Many authors have studied the influence of on finite group (see [1–9]). It is clear that a group is 1-centralizer if and only if it is abelian. In [6] Belcastro and Sherman proved that there is no -centralizer group for . They also proved that is 4-centralizer if and only if , and is 5-centralizer if and only if or . In [2] Ashrafi proved that if is 6-centralizer, then , , ？×？ ？×？ , or ？×？ ？×？ ？×？ . In [1] Abdollahi et al. proved that is 7-centralizer if and only if , or , . They also proved that if is 8-centralizer, then , or . Our main result is as follows. Theorem 1. There is no finite nonabelian group of odd order with . 2. Preliminary Results By [1], a cover for a group is a collection of proper subgroups whose union is the whole . We use the term -cover for a cover with members. A cover is called irredundant if no proper subcollection is also a cover. A cover is called a partition with kernel if the intersection of pairwise members of the cover is . Neumann in [10] obtained a uniform bound for the index of the intersection of an irredundant -cover in terms of , and Tomkinson [11] improved this bound. For a natural number , let denote the largest index , where is a group with an irredundant -cover whose intersection of all of them is . We know that , , , and (see [12–15], resp.). Now we present some lemmas and propositions that will be used in the proof of Theorem 1. Lemma 2 (Lemma 3.3 of [11]). Let be a proper subgroup of the finite group , and let , be subgroups of with and . If , then . Furthermore, if , then and for any two distinct and . Definition 3 (Definition？？2.1 of [1]). A nonempty subset of a finite group is called a set of pairwise noncommuting elements if for all distinct . A set of pairwise noncommuting elements of is said to have maximal size if its cardinality is the largest one among all such sets. Remark 4. Let be a finite group, and let be a set of