Supergroups of 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: 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.

$$A_5$$ 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
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.