In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 is said to be almost simple if it contains a non-abelian simple group

Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...

 and is contained within the automorphism group of that simple group: if it fits between a (non-abelian) simple group and its automorphism group.More precisely, a group is almost simple if it is isomorphic to such a group. In symbols, a group A is almost simple if there is a simple group S such that

Examples

• Trivially, nonabelian simple groups and the full group of automorphisms are almost simple, but proper examples exist, meaning almost simple groups that are neither simple nor the full automorphism group.
• For the symmetric group
  Symmetric group
  In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
  
    is almost simple, with the simple group being the alternating group  For is the full automorphism group of while for sits properly between and due to the exceptional outer automorphism of

Properties

The full automorphism group of a nonabelian simple group is a complete group

Complete group

In mathematics, a group G is said to be complete if every automorphism of G is inner, and the group is a centerless group; that is, it has a trivial outer automorphism group and trivial center....

 (the conjugation map is an isomorphism to the automorphism group), but proper subgroups of the full automorphism group need not be complete.

Structure

By the Schreier conjecture

Schreier conjecture

In finite group theory, the Schreier conjecture asserts that the group of outer automorphisms of every finite simple group is solvable. It was proposed by Otto Schreier in 1926, and is now known to be true as a result of the classification of finite simple groups, but no simpler proof is known....

, now generally accepted as a corollary of the classification of finite simple groups

Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...

, the outer automorphism group of a finite simple group is a solvable group

Solvable group

In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...

. Thus a finite almost simple group is an extension of a solvable group by a simple group.

External links

• Almost simple group at the Group Properties wiki