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

This article discusses some of the groups that admit the alternating group of degree five as a subgroup, quotient group, or subquotient.

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

Further information: symmetric group:S5, A5 in S5

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.

$A_{5}$ is a simple non-abelian group and $A_{5}$ and $S_{5}$ are the only two almost simple groups corresponding to $A_{5}$ .

$A_{5}$ is also of index two in the full icosahedral group, which turns out not to be $S_{5}$ , but instead the direct product of $A_{5}$ and the cyclic group of order two.

Quotients: Schur covering groups

Further information: Group cohomology of alternating group:A5#Schur multiplier, group cohomology of alternating groups, double cover of alternating group

Further information: special linear group:SL(2,5), center of special linear group:SL(2,5)

The Schur multiplier of $A_{5}$ is cyclic group:Z2.

The corresponding universal central extension (the unique Schur covering group, unique because $A_{5}$ is a perfect group) is special linear group:SL(2,5), also denoted as $2 \cdot A_{5}$ 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 $A_{5}$ .

$A_{5}$ is a simple non-abelian group and $A_{5}$ and $S L (2, 5) = 2 \cdot A_{5}$ are the only two corresponding quasisimple groups.