Supergroups of alternating group:A5
This article gives specific information, namely, supergroups, about a particular group, namely: alternating group:A5.
View supergroups of particular groups | View other specific information about alternating group:A5
Note that unlike the discussion of the subgroup structure of alternating group:A5, this discussion is necessarily not comprehensive, because there are infinitely many groups containing the alternating group of degree five.
Subgroups and quotients: essential minimalist examples
Subgroups: making all automorphisms inner
The outer automorphism group of alternating group:A5 is cyclic group:Z2, and the whole automorphism group is symmetric group:S5. Since alternating group:A5 is a centerless group, it embeds as a subgroup of index two inside its automorphism group, which is symmetric group on five elements.
Quotients: Schur covering groups
The corresponding universal central extension (the unique Schur covering group, unique because is a perfect group) is special linear group:SL(2,5), also denoted as to denote that it is a double cover (see double cover of alternating group). The center of special linear group:SL(2,5) is cyclic group:Z2 and the quotient group is .