Groups with few class sizes and the centraliser equality subgroup.
Let \(G\) be a finite \(p\)-group and let \(G\) have conjugacy class sizes \(1=n_1<n_2<\cdots<n_k\). We refer to classes of size \(n_2\) as `minimal classes', and to their elements as `minimal elements'. If \(n_s=p^{b_s}\) and \(x\in G\) has \(n_s\) conjugates, we say that \(x\) has `breadth' \(b(x)=b_s\). The `breadth' of \(G\) is \(b(G)=b_k\). Let \(D(G)=\langle x\in G\mid C_G(x)=C_G(x^p)\rangle\). The subgroup \(D(G)\) is a characteristic Abelian subgroup of \(G\). If \(G\) has \(k\) conjugacy class sizes, then \(\exp(G/D(G))\leq p^{k-1}\). (N. Ito proved this for \(k=2\).) If \(p=2\) and \(k>2\), then \(\exp(G/D(G))\leq 2^{k-2}\). We have \(D(G)\geq Z(G)\). If \(\exp(G)=p\), then \(D(G)=Z(G)\). If \(G\) is a \(2\)-group of maximal class and order \(>2^3\), then \(D(G)\) is a cyclic subgroup of index \(2\) in \(G\). If \(D(G)\leq H\leq G\) and \(\text{cl}(H)\leq p\), then \(D(G)\leq Z(H)\). Next, \(C_G(D(G))\geq Z_p(G)\), the \(p\)-th member of the upper central series of \(G\). The subgroup \(C_G(D)\) contains all normal subgroups of \(G\) of class \(<p\) and all elements of breadth \(<p\). In particular, \(C_G(D(G))\) contains all maximal Abelian normal subgroups of \(G\). The above results show the important role of the new characteristic subgroup \(D(G)\). A \(p\)-group \(G\) that is generated by its minimal classes, has class \(\leq 3\). In particular, if \(k=2\), then \(\text{cl}(G)\leq 3\) (K. Ishikawa). The subgroup, generated by the minimal classes of a \(2\)-group \(G\), is of class at most \(2\) and centralizes \(\mho_1(G)\). The subgroup, generated by the minimal classes of a metabelian \(p\)-group \(G\), is of class \(\leq 3\) and centralizes \([G,G,G]\). It is worthwhile to mention some further results presenting independent interest. Proposition 5. If \(G\) is a \(2\)-group, the minimal elements \(x\in G\) satisfying \(x^2\in Z(G)\), lie in \(Z_2(G)\). Proposition 8. Let \(B\) be an Abelian subgroup of maximal order in a metabelian \(p\)-group \(G\). Then \(G\) contains a normal Abelian subgroup \(C\leq B^G\) such that \(|C|=|B|\). Here \(B^G\) is the normal closure of \(B\) in \(G\). Lemma 9. Let \(x,y\in G\), where \(G\) is a \(p\)-group of class \(\leq p\). Then \([x^p,y]=1\) if and only if \([x,y^p]=1\). Proposition 11. Let \(N\) be a normal subgroup of a \(p\)-group \(G\) such that if two elements in \(G-N\) have distinct centralizers, then neither of these centralizers contains the other. Then either \(\exp(G/N)=p\), or \(G\) contains an Abelian maximal subgroup \(A\) and \(A\geq N\). Lemmas 12, 13 and Proposition 17. Let \(x\in G\) have breadth \(b\), and denote \(N=\langle x\rangle^G\). (a) Suppose \(o(x)=p\). Then \(N\leq p^{p^b}\). If equality holds, then \(N\) is elementary Abelian. (b) If \(x^p\in Z(G)\), then \(|N:(N\cap Z(G))|<p^{p^b}\). (c) If \(x\in G-Z(G)\), then \(\text{cl}(N)\leq b\). If \(x\in G\) with \(b(x)=b\), then \(xD(G)\in Z_{p^b}(G/D(G))\). Proposition 19. Let a group \(G\) of order \(p^n\) contain exactly \(p-1\) classes of elements of breadth \(b=b(G)\). Then \(n\leq b^2-1\). If \(b=3\), then \(p=2\), and if \(b=4\), then \(p\leq 3\). The proof of the last proposition is very involved.
scientific article; zbMATH DE number 2108127
scientific article