# Projective special linear group equals alternating group in only finitely many cases

## Contents

## Statement

The following are the cases where a Projective special linear group (?) is isomorphic to an Alternating group (?):

- is isomorphic to the alternating group on one letter, and on two letters, for any field .
- is isomorphic to the alternating group of degree four.
- and are both isomorphic to the alternating group of degree five.
- is isomorphic to the alternating group of degree six.
- is isomorphic to the alternating group of degree eight.

## Proof

### The case of degree more than two

We first consider the case where . In this case, we prove that the only solution is .

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### The case of characteristic two

We now consider the case of fields of characteristic two. In this case, the group is:

Its order is given by:

.

For this to be isomorphic to the alternating group , we must have:

.

Note that both and are both odd, so the largest power of dividing is . This yields:

.

Thus, . We thus have:

.

This puts a small bound on , namely, .as well as on , namely . A hand calculation shows that the only solutions are and . A further check shows that in this case, the groups are indeed isomorphic.

### The case of odd characteristic

We now consider the case of an odd prime, so the group is:

.

The order of the group is:

.

For this to be isomorphic to the alternating group , we get:

.

Clearly, and are relatively prime to . Thus, the largest power of dividing is . This yields:

.

Thus, we get:

.

Note that since for all primes , we get:

.

This puts a bound . A hand calculation now yields that we have the following solutions:

- .
- .
- .