In theory of finite simple groups, the following is a well celebrated

Theorem (Brauer-Fowler) If $G$ is a finite simple group and $z$ is an element of order $2$ such that centralizer of $z$ has order $n$ in $G$ then $G$ can be embedded in the alternating group $A_{n^2-1}$.

Question 1. Do any one lists an example(s) of simple group(s) in which $G$ can not be embedded in $A_{n^2-2}$?

Ignore next statements and question 2

Considering families of simple groups other than $A_n$, some simple groups in other families are isomorphic with some alternating groups (for example $\rm{PSL}_2(\mathbb{F}_5)\cong A_5$).

Question 2. Is there any example of a simple group $G$ in the family of sporadic groups or classical groups to which, by applying theorem of Brauer-Fowler, we get the embedding to be an isomorphism?

Both questions are almost similar, but I had put them in different form, concerning some special interests in the theorem stated.

  • $\begingroup$ With your note, the question 2 has no good meaning. I will put [ignore] on it. $\endgroup$ – p Groups Oct 3 '16 at 6:22
  • $\begingroup$ @Derek: wait; $GL_4(2)$ is $A_8$, isn't it? And now question 2. In $GL_4(2)$, is there any involution whose centralizer has order $3$? (Then I can apply theorem of Brauer to get injection from $GL_4(2)$ to $A_8$, and because of same orders, the embedding is isomorphism.) $\endgroup$ – p Groups Oct 3 '16 at 6:25
  • $\begingroup$ Sorry, I'm going senile! Let's start again. The examples of alternating groups isomorphic to finite simple groups in other families are $A_5 \cong {\rm PSL}(2,4) \cong {\rm PSL}(2,5)$, $A_6 \cong {\rm PSL}(2,9)$, and $A_8 \cong {\rm PSL}(4,2)$. $\endgroup$ – Derek Holt Oct 3 '16 at 8:21
  • $\begingroup$ No involution could possibly have a centralizer of order $3$. There are two conjugacy classes of involutions in ${\rm GL}_4(2)$, and their centralizers have orders $192$ and $96$. $\endgroup$ – Derek Holt Oct 3 '16 at 8:27

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