# Simple non-abelian group is isomorphic to subgroup of alternating group on left coset space of proper subgroup of finite index

From Groupprops

## Statement

Suppose is a simple non-abelian group and is a proper subgroup of that is a subgroup of finite index in . Then, is isomorphic to a subgroup of the alternating group on the left coset space .

## Facts used

- Simple non-abelian group is isomorphic to subgroup of alternating group on left coset space of proper subgroup
- Normality satisfies transfer condition: The intersection of a normal subgroup and any subgroup is normal in the latter subgroup.
- Second isomorphism theorem

## Proof

**Given**: A simple non-Abelian group , a proper subgroup of finite index in .

**To prove**: is isomorphic to a subgroup of the alternating group on .

**Proof**:

- By fact (1), is isomorphic to a subgroup, say , of .
- By definition, the alternating group, , is normal in . Thus, by fact (2), is normal in .
- Since is simple, or intersects trivially. We consider both cases:
- : In this case, , so is isomorphic to a subgroup, , of .
- is trivial: In this case, the second isomorphism theorem (fact (3)) yields that . The left side is a subgroup of , which is a group of order two. But a group of order two has no simple non-Abelian subgroups, so this case is not possible.