A new characterization of the automorphism groups of Mathieu groups

Let cd ( G ) {\rm{cd}}\left(G) be the set of irreducible complex character degrees of a finite group G G . ρ ( G ) \rho \left(G) denotes the set of primes dividing degrees in cd ( G ) {\rm{cd}}\left(G) . For any prime p, let p e p ( G ) = max { χ ( 1 ) p ∣ χ ∈ Irr ( G ) } {p}^{{e}_{p}\left(G)}=\max \left\{\chi {\left(1)}_{p}\hspace{0.08em}| \hspace{0.08em}\chi \in {\rm{Irr}}\left(G)\right\} and V ( G ) = { p e p ( G ) ∣ p ∈ ρ ( G ) } V\left(G)=\left\{{p}^{{e}_{p}\left(G)}\hspace{0.08em}| \hspace{0.1em}p\in \rho \left(G)\right\} . The degree prime-power graph Γ ( G ) \Gamma \left(G) of G G is a graph whose vertices set is V ( G ) V\left(G) , and two vertices x , y ∈ V ( G ) x,y\in V\left(G) are joined by an edge if and only if there exists m ∈ cd ( G ) m\in {\rm{cd}}\left(G) such that x y ∣ m xy| m . It is an interesting and difficult problem to determine the structure of a finite group by using its degree prime-power graphs. Qin proved that all Mathieu groups can be uniquely determined by their orders and degree prime-power graphs. In this article, we continue this topic and successfully characterize all the automorphism groups of Mathieu groups by using their orders and degree prime-power graphs.

Locally projective graphs and their densely embedded subgraphs

The article contributes to the classification project of locally projective graphs and their locally projective groups of automorphisms outlined in Chapter 10 of Ivanov (The Mathieu Groups, Cambridge University Press, Cambridge, 2018). We prove that a simply connected locally projective graph $$\Gamma $$ Γ of type (n, 3) for $$n \ge 3$$ n ≥ 3 contains a densely embedded subtree provided (a) it contains a (simply connected) geometric subgraph at level 2 whose stabiliser acts on this subgraph as the universal completion of the Goldschmidt amalgam $$G_3^1\cong \{S_4 \times 2,S_4 \times 2\}$$ G 3 1 ≅ { S 4 × 2 , S 4 × 2 } having $$S_6$$ S 6 as another completion, (b) for a vertex x of $$\Gamma $$ Γ the group $$G_{\frac{1}{2}}(x)$$ G 1 2 ( x ) which stabilizes every line passing through x induces on the neighbourhood $$\Gamma (x)$$ Γ ( x ) of x the (dual) natural module $$2^n$$ 2 n of $$G(x)/G_{\frac{1}{2}}(x) \cong L_n(2)$$ G ( x ) / G 1 2 ( x ) ≅ L n ( 2 ) , (c) G(x) splits over $$G_{\frac{1}{2}}(x)$$ G 1 2 ( x ) , (d) the vertex-wise stabilizer $$G_1(x)$$ G 1 ( x ) of the neighbourhood of x is a non-trivial group, and (e) $$n \ne 4$$ n ≠ 4 .

Cohomology of Groups

This chapter introduces the basic ingredients of the cohomology of groups and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: integral homology of finite groups such as the Mathieu groups, homology of crystallographic groups, homology of nilpotent groups, homology of Coxeter groups, transfer homomorphism, homological perturbation theory, mod-p comology rings of small finite p-groups, Lyndon-Hocshild-Serre spectral sequence, Bokstein operation, Steenrod squares, Stiefel-Whitney classes, Lie algebras, the modular isomorphism problem, and Bredon homology.