A characterization of the Janko group \(J_1\) by an active fragment of its character table (Q2770605)

Let \(G\) be a finite group and let \(D\) be a normal subset of \(G\). Let \(\alpha\) be a complex-valued class function on \(G\). Define a new class function \(\alpha|^0_D\) by \(\alpha|^0_D(g)=\alpha(g)\) if \(g\in D\) and \(\alpha|^0_D(g)=0\) otherwise. Let \(\Phi\) be a non-empty subset of irreducible complex characters of \(G\). We say that \(D\) and \(\Phi\) interact if, for each \(\phi\in\Phi\), \(\phi|^0_D\) is a linear combination (with complex coefficients) of characters in \(\Phi\). That portion of the character table of \(G\) labelled by the characters in \(\Phi\) and the conjugacy classes in \(D\) is called an active fragment of the character table. This definition is due to the author.NEWLINENEWLINENEWLINEThe author points out that the smallest Janko group \(J_1\) has a \(12\times 6\) active fragment in its character table. The corresponding \(\Phi\) consists of the principal character, two characters of degree \(56\), two of degree \(76\), three of degree \(77\), three of degree \(133\), and one of degree \(209\). \(D\) is the union of the identity, two conjugacy classes consisting of elements of order \(7\) and \(11\), respectively, and three conjugacy classes consisting of elements of order \(19\). His main theorem is that such an active fragment characterizes \(J_1\). He makes substantial use of theorems relating to active fragments previously proved by himself. He establishes the group order and proves that the group is simple. Ultimately, classification by the centralizer of an involution is used.