Examples of Infinite Simple Groups

Question

I would like a list of infinite simple groups. I am only aware of $A_\infty$.

Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that $PSL_n(F)$ is simple.

References about this topic, or any example, are also appreciated.

Answer 1

Suppose that you have a proper chain of inclusions of nonabelian simple groups $$G_1 \subsetneq G_2 \subsetneq G_3 \subsetneq \cdots.$$ Then $\bigcup_{i = 1}^\infty G_i$ is an infinite nonabelian simple group.

The group $A_\infty$ is the union of the chain $A_5 \subset A_6 \subset A_7 \subset \cdots$ of finite alternating groups.

Answer 2

Richard Thompson's groups $T$ and $V$ are well-known examples of infinite simple groups. See this answer of mine for more details, or look up the article Introductory notes on Richard Thompson's groups by Cannon, Floyd and Parry. They are defined by their action on the unit circle.

Answer 3

A Tarski monster group is a finitely generated, infinite group where every proper, non-trivial subgroup is cyclic of order a fixed prime $p$. These were shown to exist for all $p>>1$ in the 80s by Ol'shanskii. Moreover, they are simple groups.

To see that Tarski monster groups are simple, suppose $N$ is a normal subgroup of a Tarski monster group $G$. Then pick some proper subgroup $M\neq N$. As $N$ is normal, $MN$ is a subgroup of order $p^2$, a contradiction.

Answer 4

Here is the list of examples we have which were directed by the comments thus far:

• $PSL_n(K)$ for when $K$ is an infinite field and $n\geq 2$ ¹
• The finitary alternating group $A(\kappa)$ for any infinite cardinal $\kappa$ ²

REFERENCES

1. This entry at Groupprops shows that $PSL_n(K)$ is actually simple for all $n\geq 2$ and any $K$ except for $PSL_2(\Bbb F_2)$ and $PSL_2(\Bbb F_3)$
2. This question addresses this fact