Supergroups of 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: 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 $SL(2,5) = 2 \cdot A_5$ are the only two corresponding quasisimple groups.