What is the second least non prime order of a simple group?

Question

Let $G$ be a simple group such that $|G|$ is not a prime.

I have shown that $|G|\geq 60$ and there is a simple group of order $60$. (Namely, $A_5$)

Informally speaking, this means that the first simple group is of order $60$.

What is the second one?

Since $A_6$ is simple, the second one should be $\leq 360$, but I am not sure whether there is no simple group between $60$ and $360$

score 2 · Accepted Answer · answered Sep 29 '14 at 09:30

2

The second one is the projective special linear group $PSL(2,7)$ of order $168$. With $168$ elements it is indeed the second-smallest nonabelian simple group after the alternating group $A_5$ with 60 elements. It has been studied a lot. For a first reference see here. The third one then is $A_6$ of order $360$.

answered Sep 29 '14 at 09:30

Dietrich Burde

130,978

PSL(3,2), not PSL(7,2) – Dima Pasechnik Sep 29 '14 at 09:31
1

No, I mean $PSL(2,7)$ (which is isomorphic to your groups, since there is one one simple nonabelian group of order $168$ up to isomorphism). – Dietrich Burde Sep 29 '14 at 09:33

score 0 · Answer 2 · answered Sep 29 '14 at 09:31

0

The next nonabelian simple group has order 168. It can be described as the group of invertible 3x3 matrices mod 2, and usually denoted $GL_3(2)$, or $PSL_3(2)$.

answered Sep 29 '14 at 09:31

Dima Pasechnik

445

What is the second least non prime order of a simple group?

2 Answers2