Almost simple group

It is said that a group is almost simple if it contains a non - Abelian simple group and is contained in the automorphism group of this simple group. In a character notation, a group A is almost simple if there is a simple group S such that $S\leq A\leq \operatorname {Aut} (S)$ ${\ displaystyle S \ leq A \ leq \ operatorname {Aut} (S)}$ ${\ displaystyle S \ leq A \ leq \ operatorname {Aut} (S)}$ ^[1] .

Content

Examples

Trivially, non-Abelian simple groups and complete automorphism groups are almost simple, but there are examples of almost simple groups that are neither simple nor complete automorphism groups.
For $n=5$ ${\ displaystyle n = 5}$ or $n\geqslant 7$ ${\ displaystyle n \ geqslant 7}$ symmetric group $S_{n}$ ${\ displaystyle S_ {n}}$ is a group of automorphisms of a simple alternating group $A_{n},$ ${\ displaystyle A_ {n},}$ so that $S_{n}$ ${\ displaystyle S_ {n}}$ is almost simple in this trivial sense.
For $n=6$ ${\ displaystyle n = 6}$ there is a pure example since $S_{6}$ ${\ displaystyle S_ {6}}$ is purely between a simple group $A_{6}$ ${\ displaystyle A_ {6}}$ and $\operatorname {Aut} (A_{6}),$ ${\ displaystyle \ operatorname {Aut} (A_ {6}),}$ due to group $A_{6}$ ${\ displaystyle A_ {6}}$ . Two other groups, Mathieu Group $M_{10}$ ${\ displaystyle M_ {10}}$ and projective full linear group $\operatorname {PGL} _{2}(9)$ ${\ displaystyle \ operatorname {PGL} _ {2} (9)}$ are also purely between $A_{6}$ ${\ displaystyle A_ {6}}$ and $\operatorname {Aut} (A_{6}).$ ${\ displaystyle \ operatorname {Aut} (A_ {6}).}$

Properties

The automorphism group of a non-Abelian simple group is a complete group (the map of adjacent classes is an isomorphism into an automorphism group), but the proper subgroup of a complete automorphism group is not necessarily complete.

Structure

According to , now universally accepted as a consequence of the classification of simple finite groups , the finite simple group is a solvable group ^[2] . Thus, a finite simple group is an extensible soluble group over a simple group.

Notes

↑ Vdovin, 2007 , p. 159.
↑ Vdovin, Revin, 2011 , p. eleven.

Literature

Vdovin E.P. Carter subgroups in finite almost simple groups // Algebra and Logic. - 2007.- T. 46 , no. 2 . - S. 157–216 .
Vdovin E.P., Revin D.O. Sylow type theorems // SUCCESSES OF MATHEMATICAL SCIENCES. - 2011 .-- T. 66 , no. 5 (401) . - S. 3–46 .

Links

Almost simple group at the Group Properties wiki

[_fdc76da173d5f6c0-1] Vdovin, 2007 , p. 159.

[_b4db234977152fac-2] Vdovin, Revin, 2011 , p. eleven.