Supergroups of special linear group:SL(2,5)

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This article gives specific information, namely, supergroups, about a particular group, namely: special linear group:SL(2,5).
View supergroups of particular groups | View other specific information about special linear group:SL(2,5)

This article discusses some of the groups that admit special linear group:SL(2,5) as a subgroup, quotient group, or subquotient.

Note that unlike the discussion of subgroup structure of special linear group:SL(2,5), the discussion is necessarily not comprehensive, because there are infinitely many groups containing any given group.

Subgroups and quotients: essential minimalist examples

Subgroups: making all automorphisms inner

The automorphism group of SL(2,5) is isomorphic to symmetric group:S5, which can be more explicitly thought of as PGL(2,5) in this context. Explicitly, the elements of GL(2,5) act as automorphisms of SL(2,5) by conjugation, and the kernel of this action is the center, so this induces a faithful action of PGL(2,5) on SL(2,5) and an embedding of PGL(2,5) inside the automorphism group of SL(2,5). In this case, that embedding is surjective, i.e., SL(2,5) has no further automorphisms.

There are a number of ways we can find a group G containing a subgroup H isomorphic to SL(2,5) as a normal subgroup so that the induced action by conjugation yields a surjective homomorphism G/C_G(H) \to \operatorname{Aut}(H) (in other words, G contains a normal fully normalized subgroup isomorphic to SL(2,5)).

The minimal examples are those where H is a self-centralizing subgroup of G (i.e., C_G(H) = Z(H)), i.e., is a NSCFN-subgroup of G. In this case, G has normal subgroup Z(H) of order two, with quotient group \operatorname{Aut}(H) of order 120. In particular, G is a double cover of \operatorname{Aut}(H) \cong S_5. In particular, G must have order 240.

The upshot is that the classification of groups that contain SL(2,5) as a NSCFN-subgroup is exactly the same as the classification of double covers of the automorphism group, which is isomorphic to symmetric group:S5. There are two such groups, both of order 240:

For more, see group cohomology of symmetric groups, double cover of symmetric group, group cohomology of symmetric group:S5, and supergroups of symmetric group:S5.

In addition, there are many non-minimal examples. For instance, general linear group:GL(2,5) is a group of order 480 containing SL(2,5) as a normal fully normalized subgroup of index four.

Quotients: Schur covering groups

The group special linear group:SL(2,5) is a Schur-trivial group, i.e., its Schur multiplier is a trivial group. Further, the group is a perfect group. Thus, its Schur covering group is itself.