We associate a graph N G with a group

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For any group G, let |Cent(G)| denote the number of centralizers of its elements. A group G is called n-centralizer if |Cent(G)| = n. In this paper, we find |Cent(G)| for all minimal simple groups. Using these results we prove that there exist finite simple groups G and H with the property that |Cent(G)| = |Cent(H)| but G ∼ = H. This result gives a negative answer to a question raised by A. Ashrafi and B. Taeri. We also characterize all finite semi-simple groups G with |Cent(G)| ≤ 73.

For any group G, let C(G) denote the set of centralizers of G. We say that a group G has n centralizers (G is a Cn-group) if |C(G)| = n.In this note, we show that the derived length of a soluble Cn-group (not necessarily finite) is bounded by a function of n.Mathematics Subject Classification (2010). Primary 20E07; Secondary 20D10.

A subset $X$ of a group $G$ is a set of pairwise noncommuting elements if $ab\neq ba$ for any two distinct elements $a$ and $b$ in $X$. If $|X|\geq |Y|$ for any other set of pairwise noncommuting elements $Y$ in $G$, then $X$ is called a maximal subset of pairwise noncommuting elements and the cardinality of such a subset (if it exists) is denoted by ${\it\omega}(G)$. In this paper, among other things, we prove that, for each positive integer $n$, there are only finitely many groups $G$, up to isoclinism, with ${\it\omega}(G)=n$, and we obtain similar results for groups with exactly $n$ centralisers.