Quasisimple group

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In

${\displaystyle 1\to Z(E)\to E\to S\to 1}$

such that ${\displaystyle E=[E,E]}$, where ${\displaystyle Z(E)}$ denotes the center of E and [ , ] denotes the commutator.^[1]

Equivalently, a group is quasisimple if it is equal to its commutator subgroup and its inner automorphism group Inn(G) (its quotient by its center) is simple (and it follows Inn(G) must be non-abelian simple, as inner automorphism groups are never non-trivial cyclic). All non-abelian simple groups are quasisimple.

The

The quasisimple groups are often studied alongside the simple groups and groups related to their automorphism groups, the almost simple groups. The representation theory of the quasisimple groups is nearly identical to the projective representation theory of the simple groups.

Examples

The covering groups of the alternating groups are quasisimple but not simple, for ${\displaystyle n\geq 5.}$

See also

• Almost simple group
• Schur multiplier
• Semisimple group

References

• Aschbacher, Michael (2000). Finite Group Theory.
  Zbl 0997.20001
  .

• http://mathworld.wolfram.com/QuasisimpleGroup.html

Notes

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