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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.

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    Is $A_\infty$ the finitary alternating group on a set of countable cardinality? – superAnnoyingUser Jun 22 '14 at 06:54
  • @Student yes it is –  Jun 22 '14 at 07:19
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    I am confused, because you say that you are only aware of $A_\infty$, but then proceed to mention ${\rm PSL}_n(F)$ for infinite fields $F$ (with $n>1$) which provide further examples. – Derek Holt Jun 22 '14 at 09:36
  • @DerekHolt I'm not sure if such examples exist (I'm just most interested in them if they do). –  Jun 22 '14 at 10:07
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    There are not just examples of such fields and such $n$. All $n \geq 2$ and all infinite fields (also most finite fields) work. – Tobias Kildetoft Jun 22 '14 at 10:13
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    The same is true for the other classical groups and untwisted groups of Lie type (the only exceptional fields are small). -- A completely different type of simple group is given by the Tarski monsters. – Jack Schmidt Jun 22 '14 at 17:33
  • @JackSchmidt I'm having trouble finding information on this (I got the Tarski monsters). Can you elaborate? –  Jun 23 '14 at 13:50
  • http://en.wikipedia.org/wiki/Classical_group -- If $G$ is a classical group (over any field where they are defined), then $[G,G]/(Z(G) \cap [G,G])$ is simple except for a few tiny fields. These are things like orthogonal groups, symplectic groups, and unitary groups. There are some other matrix groups too (G2, F4, E6, E7, E8) that have similar properties. Every finite simple group is one of those, alternating, sporadic, or twisted versions of those (2An = unitary, 2Dn = more orthogonal, 3D4, 2E6 and the very twisted 2B2, 2G2, 2F4). – Jack Schmidt Jun 23 '14 at 15:11
  • Short version: PSL and Alt are some of the finite simple groups. They have infinite versions. There are other finite simple groups. All but the sporadics have infinite versions. – Jack Schmidt Jun 23 '14 at 15:12

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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.

user1729
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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.

user26857
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spin
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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
Community
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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.

Community
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user1729
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a.) As mentioned above, the projective special linear group $\operatorname{PSL}_n (F)$ is infinite and simple for any infinite field $F$ and any integer $n>1$. In fact, $\operatorname{PSL}_n(F)$ is simple if $n\geq 3$ for any field $F$ and if $n=2$ and $|F|\geq 4$.

b.) The group $\operatorname{SO}_3(\mathbb{R})$ (which may be seen as a group of rotations of a sphere $\mathbb{S}^2$) is simple (and obviously infinite).

c.) The projective symplectic group over an infinite field $F$ $$\operatorname{PSp}_{2n}(F):=\frac{\operatorname{Sp}_{2n}(F)}{Z(\operatorname{Sp}_{2n}(F))}$$ is simple. In fact, $\operatorname{PSp}_{2n}(F)$ is siple for any $n\geq 1$ and $F$ except the cases $n=1 \ \text{and} \ F = \mathbb{F}_2$, $n=1 \ \text{and} \ F = \mathbb{F}_3$, $n=2 \ \text{and} \ F = \mathbb{F}_2$.

Further reading:

1.) https://cameroncounts.wordpress.com/wp-content/uploads/2013/12/cg.pdf Here you may know more about examples a.) and c.) (despite the fact that this book is focused on "classical finite groups").

Mitya Kustov
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