## FANDOM

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A group of finite Morley rank is a group $(G,\cdot)$, usually with extra structure, whose Morley rank is less than $\omega$.

The Cherlin-Zilber conjecture asserts that every simple group of finite Morley rank is an algebraic group over a field. This remains open as of 2014.

However, a considerable amount is known about groups of finite Morley rank. See for example, Bruno Poizat's book Stable Groups, as well as…[more recent books]

• Morley rank and Lascar rank coincide, and are definable. In particular, Morley rank satisfies the Lascar inequalities.
• If $G$ is a group of finite Morley rank, then the connected component $G^0$ exists, and is definable, rather than merely being type-definable. There is a unique type in $G^0$ of maximal Morley rank, i.e., $G^0$ has Morley degree 1. The translates of $G^0$ are called the generics of $G$, and have many good properties. They are the unique types which are translation invariant.
• Any field of finite Morley rank is algebraically closed, but may have additional structure.
• A group of finite Morley rank is simple (in the group theoretic sense) if and only if it is definable simple. That is, if $G$ is not simple as an abstract group, then $G$ has a definable normal subgroup.
• Every infinite group of finite Morley rank contains an infinite abelian definable subgroup.
• Every simple group of finite Morley rank is almost strongly minimal, i.e., is algebraic over a strongly minimal set.
• Groups of finite Morley rank are "dimensional." This falls out of the Lascar analysis.
• Every type-definable subgroup of a group of finite Morley rank is, in fact, definable.

## Transitive action on a strongly minimal set Edit

One rather strong result about groups of finite Morley rank is the following:

Let $G$ be a group of finite Morley rank, acting transitively and faithfully on a strongly minimal set $S$. Then we are in one of the following three situations:

• $G$ has rank 1, is commutative, and $S$ is a $G$-torsor.
• $G$ has rank 2, $S$ is the affine line over a definable field $K$, and $G$ is the group of affine linear transformations over $K$
• $G$ has rank 3, $G$ is $PSL_2(K)$ for a definable field $K$, and $S$ is the projective line over $K$, with the usual action.

In cases 2 or 3, $K$ is algebraically closed. $G$ cannot have rank greater than 3.

Under the hypothesis that there are no bad groups, it can be shown that this implies that the Cherlin-Zilber conjecture holds for groups of Morley rank at most 3: any simple group of Morley rank at most 3 must be $PSL_2(K)$ for a definable field $K$.

It also implies that if $G$ is a simple group of finite Morley rank, containing a definable subgroup $H$ such that $RM(H) = RM(G) - 1$, then $G$ has rank 3 and is $PSL_2(K)$ over an algebraically closed definable field.