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Jordan’s theorem is a basic theorem in the theory of finite linear groups, and can be formulated as follows:

Theorem 1 (Jordan’s theorem) Let ${G}$ be a finite subgroup of the general linear group ${GL_d({\bf C})}$. Then there is an abelian subgroup ${G'}$ of ${G}$ of index ${[G:G'] \leq C_d}$, where ${C_d}$ depends only on ${d}$.

Informally, Jordan’s theorem asserts that finite linear groups over the complex numbers are almost abelian. The theorem can be extended to other fields of characteristic zero, and also to fields of positive characteristic so long as the characteristic does not divide the order of ${G}$, but we will not consider these generalisations here. A proof of this theorem can be found for instance in these lecture notes of mine.

I recently learned (from this comment of Kevin Ventullo) that the finiteness hypothesis on the group ${G}$ in this theorem can be relaxed to the significantly weaker condition of periodicity. Recall that a group ${G}$ is periodic if all elements are of finite order. Jordan’s theorem with “finite” replaced by “periodic” is known as the Jordan-Schur theorem.

The Jordan-Schur theorem can be quickly deduced from Jordan’s theorem, and the following result of Schur:

Theorem 2 (Schur’s theorem) Every finitely generated periodic subgroup of a general linear group ${GL_d({\bf C})}$ is finite. (Equivalently, every periodic linear group is locally finite.)

Remark 1 The question of whether all finitely generated periodic subgroups (not necessarily linear in nature) were finite was known as the Burnside problem; the answer was shown to be negative by Golod and Shafarevich in 1964.

Let us see how Jordan’s theorem and Schur’s theorem combine via a compactness argument to form the Jordan-Schur theorem. Let ${G}$ be a periodic subgroup of ${GL_d({\bf C})}$. Then for every finite subset ${S}$ of ${G}$, the group ${G_S}$ generated by ${S}$ is finite by Theorem 2. Applying Jordan’s theorem, ${G_S}$ contains an abelian subgroup ${G'_S}$ of index at most ${C_d}$.

In particular, given any finite number ${S_1,\ldots,S_m}$ of finite subsets of ${G}$, we can find abelian subgroups ${G'_{S_1},\ldots,G'_{S_m}}$ of ${G_{S_1},\ldots,G_{S_m}}$ respectively such that each ${G'_{S_j}}$ has index at most ${C_d}$ in ${G_{S_j}}$. We claim that we may furthermore impose the compatibility condition ${G'_{S_i} = G'_{S_j} \cap G_{S_i}}$ whenever ${S_i \subset S_j}$. To see this, we set ${S := S_1 \cup \ldots \cup S_m}$, locate an abelian subgroup ${G'_S}$ of ${G_S}$ of index at most ${C_d}$, and then set ${G'_{S_i} := G'_S \cap G_{S_i}}$. As ${G_S}$ is covered by at most ${C_d}$ cosets of ${G'_S}$, we see that ${G_{S_i}}$ is covered by at most ${C_d}$ cosets of ${G'_{S_i}}$, and the claim follows.

Note that for each ${S}$, the set of possible ${G'_S}$ is finite, and so the product space of all configurations ${(G'_S)_{S \subset G}}$, as ${S}$ ranges over finite subsets of ${G}$, is compact by Tychonoff’s theorem. Using the finite intersection property, we may thus locate a subgroup ${G'_S}$ of ${G_S}$ of index at most ${C_d}$ for all finite subsets ${S}$ of ${G}$, obeying the compatibility condition ${G'_T = G'_S \cap G_T}$ whenever ${T \subset S}$. If we then set ${G' := \bigcup_S G'_S}$, where ${S}$ ranges over all finite subsets of ${G}$, we then easily verify that ${G'}$ is abelian and has index at most ${C_d}$ in ${G}$, as required.

Below I record a proof of Schur’s theorem, which I extracted from this book of Wehrfritz. This was primarily an exercise for my own benefit, but perhaps it may be of interest to some other readers.

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