On the uniqueness of the Harada-Norton group
By definition, \(2\cdot HS\) denotes the non-split extension of the group of order 2 by the Higman-Sims sporadic finite simple group and \((2\cdot HS):2\) denotes the split extension of \((2\cdot HS)\) by a group of order 2 with center of order 2. Also \(2^{1+8}_ +\) denotes an extraspecial group of order \(2^ 9\) and of type + and a group \(H\) is of \(2^{1+8}_ +\cdot(A_ 5\text{ wr }2)\)-type if \(H\) is 2-constrained and a nonsplit extension of \(2^{1+8}_ +\) by \((A_ 5\text{ wr }2)\). A finite group \(G\) is said to be of \(F_ 5\) type (where \(F_ 5\) denotes a Harada- Norton simple group) if \(G\) contains involutions \(a\), \(z\) such that \(C_ G(a)\cong (2\cdot HS):2\) and \(C_ G(z)\) is of \(2^{1+8}_ +\cdot(A_ 5\text{ wr }2)\)-type. The existence of the Monster implies the existence of a group of \(F_ 5\)-type. The main result of this paper establishes the uniqueness of the Harada-Norten simple group \(F_ 5\): Main Theorem: Let \(G\) be a finite group of \(F_ 5\)-type. Then \(G\) is unique up to isomorphism. Let \(G\) be a finite group of \(F_ 5\)-type. Consider the graph \(\Gamma\): vertex set is \(a^ G\) and \(x,y\in a^ G\) form an edge if \(xy \in a^ G\). The main theorem is established by demonstrating the uniqueness of \(\Gamma\) and that \(G=[\text{Aut}(\Gamma),\text{Aut}(\Gamma)]\).
Higman-Sims sporadic finite simple group
scientific article; zbMATH DE number 165838