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Published in:
2010 | OriginalPaper | Chapter
Free Profinite Groups
Authors : Luis Ribes, Pavel Zalesskii
Published in: Profinite Groups
Publisher: Springer Berlin Heidelberg
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Let
$\mathcal{N}$
be a nonempty collection of normal subgroups of finite index of a group
G
and assume that
$\mathcal{N}$
is filtered from below, i.e.,
$\mathcal{N}$
satisfies the following condition:
$$\mbox{whenever }N_1,N_2\in \mathcal{N},\mbox{ there exists }N\in \mathcal{N}\mbox{ such that }N\le N_1\cap N_2.$$
Then one can make
G
into a topological group by considering
$\mathcal{N}$
as a fundamental system of neighborhoods of the identity element 1 of
G
. We refer to the corresponding topology on
G
as a
profinite topology
. If every quotient
G
/
N
$(N\in \mathcal{N})$
belongs to a certain class
$\mathcal{C}$
, we say more specifically that the topology above is
a pro -
$\mathcal{C}$
topology
.