Recognizing direct products from their conjugate type vectors
Let \(x\) be an element of a finite group \(G\); then \(|G:C_G(x)|\) is called the index of \(x\) in \(G\). Let \(n_r>\cdots>n_2>n_1=1\) denote all the distinct indices of elements of \(G\). Then \((n_r,\dots,n_2,n_1)\) is called the conjugate type vector of \(G\). If, in addition, \((m_s,\dots,m_2,m_1)\) is the conjugate type vector of \(H\), then \((n_r,\dots,n_2,n_1)\times(m_s,\dots,m_2,m_1)=(n_im_j\mid 1\leq i\leq r,\;1\leq j\leq s)\) is, after reordering of members, the conjugate type vector of \(G\times H\). Main result of this note is the following Theorem 1. Suppose \(G\) has conjugate type vector \((p^{a_s}_s,\dots,p^{a_1}_1,1)\times(q^{b_r}_r,\dots,q^{b_1}_1,1)\), where \(p_s,\dots,p_1\), \(q_r,\dots,q_1\) are distinct primes. Then \(r,s\leq 2\) and \(G=A\times B\), where \(A\) has conjugate type vector \((p^{a_s}_s,\dots,p^{a_1}_1,1)\) and \(B\) has conjugate type vector \((q^{b_r}_r,\dots,q^{b_1}_1,1)\).
scientific article; zbMATH DE number 1565875