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Summary: Symmetric groups as products of Abelian
subgroups
Miklos Abert y
Abstract
We prove that the full symmetric group over any in nite set is the
product of nitely many Abelian subgroups. In fact, 289 subgroups suÆce.
We also obtain sharp bounds on the minimal number k such that the
nite symmetric group Sn is the product of k Abelian subgroups. Using
this, we prove that Sn is the product of 72n 1=2 (log n) 3=2 cyclic subgroups.
1 Introduction
A group G is said to have nite Abelian width if G is a product of nitely many
Abelian subgroups, i.e., G = A 1 A 2 A k = fa 1 a 2 a k j a i 2 A i g for some
Abelian subgroups A 1 , A 2 , : : : , A k . This notion was introduced in [2] where
it is shown that for an arbitrary eld K the special linear group SL(n; K) is a
product of 60 Abelian subgroups.
The related class of groups having nite cyclic width (de ned analogously)
has been much investigated (see [6], [8] and [10]).
Here we rst prove the following somewhat surprising result.
Theorem 1. The full symmetric group over any in nite set is a product of
nitely many Abelian subgroups.
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