Characterization of automorphism groups of sporadic simple groups

It is proved that all automorphism groups of the sporadic simple groups are characterized by their element orders and the group orders.

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group,π_e(G)denotes the set of orders of elements in G.If Ω is a subset of the set of natural numbers,h(Ω)stands for the number of isomorphism classes of finite groups with the same set Ω of element orders.We say that G is k-distinguishable if h(π_(G))=k<∞,otherwise G is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.It is shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M^cL and O'N, then Aut(M)is charaeterizable by its dement orders.It is also proved that if M is isomorphic to M_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}.

Recognizing by Spectrum for the Automorphism Groups of Sporadic Simple Groups

The spectrum of a finite group is the set of its element orders,and two groups are said to be isospectral if they have the same spectra.A finite group G is said to be recognizable by spectrum,if every finite group isospectral with G is isomorphic to G.We prove that if S is one of the sporadic simple groups M^(c)L,M_(12),M_(22),He,Suz and O'N,then Aut(S)is recognizable by spectrum.This finishes the proof of the recognizability by spectrum of the automorphism groups of all sporadic simple groups,except J_(2).

Flag-transitive 2-(v,5,λ)designs with sporadic socle

We state that the flag-transitive automorphism group of a 2-(v,5,λ)design D is primitive of affine type or almost simple type.W e also find that there are up to isomorphism 202-(v,5,λ)designs admitting flag-transitive automorphism groups with socle of a sporadic simple group.