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In mathematics, a group is said to be almost simple if it contains a non-abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) simple group and its automorphism group. In symbols, a group ${\displaystyle A}$ is almost simple if there is a (non-abelian) simple group S such that ${\displaystyle S\leq A\leq \operatorname {Aut} (S)}$, where the inclusion of ${\displaystyle S}$ in ${\displaystyle \mathrm {Aut} (S)}$ is the action by conjugation, which is faithful since ${\displaystyle S}$ is has trivial center.

Examples

• Trivially, non-abelian simple groups and the full group of automorphisms are almost simple. For ${\displaystyle n=5}$ or ${\displaystyle n\geq 7,}$ the symmetric group ${\displaystyle \mathrm {S} _{n}}$ is the automorphism group of the simple alternating group ${\displaystyle \mathrm {A} _{n},}$ so ${\displaystyle \mathrm {S} _{n}}$ is almost simple in this trivial sense.
• For ${\displaystyle n=6}$ there is a proper example, as ${\displaystyle \mathrm {S} _{6}}$ sits properly between the simple ${\displaystyle \mathrm {A} _{6}}$ and ${\displaystyle \operatorname {Aut} (\mathrm {A} _{6}),}$ due to the exceptional outer automorphism of ${\displaystyle \mathrm {A} _{6}.}$ Two other groups, the Mathieu group ${\displaystyle \mathrm {M} _{10}}$ and the projective general linear group ${\displaystyle \operatorname {PGL} _{2}(9)}$ also sit properly between ${\displaystyle \mathrm {A} _{6}}$ and ${\displaystyle \operatorname {Aut} (\mathrm {A} _{6}).}$

Properties

The full automorphism group of a non-abelian simple group is a complete group (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, now generally accepted as a corollary of the classification of finite simple groups, the outer automorphism group of a finite simple group is a solvable group. Thus a finite almost simple group is an extension of a solvable group by a simple group.

See also

• Quasisimple group
• Semisimple group

Notes

External links

• Almost simple group at the Group Properties wiki

1. Dallavolta, F.; Lucchini, A. (1995-11-15). "Generation of Almost Simple Groups". Journal of Algebra. 178 (1): 194–223. doi:10.1006/jabr.1995.1345. ISSN 0021-8693.
2. Robinson, Derek J. S. (1996), Robinson, Derek J. S. (ed.), "Subnormal Subgroups", A Course in the Theory of Groups, Graduate Texts in Mathematics, vol. 80, New York, NY: Springer, Corollary 13.5.10, doi:10.1007/978-1-4419-8594-1_13, ISBN 978-1-4419-8594-1, retrieved 2024-11-23

• Properties of groups